£)&?)$ 

Preprinted  from  the  January,  February,  March  and  April 
numbers  of  Yol.  XVIII,  1918,  School  Science 
and  Mathematics 

THE  STATUS  OF  MATHEMATICS  IN  SECONDARY 
SCHOOLS1. 

r 

By  Alfred  Davis, 

Francis  W.  Parker  School,  330  Webster  Ave.,  Chicago. 
Chairman  of  a  Committee  Appointed  by  the  Mathematics  Club 

of  Chicago. 

■  . 

One  might  judge  by  the  criticisms  that  come  from  certain 
quarters  that  algebra  and  geometry  are  in  a  precarious  situa¬ 
tion.  Some  would-be  reformers  would  have  us  abandon  the 
teaching  of  these  subjects  in  high  school  in  favor  of  what  are 
said  to  be  more  useful  subjects.  Others  would  make  them 
optional  studies,  or  they  would  devitalize  them  by  making  them 
an  easy  and  pleasant  pastime  for  all  who  are  required  to  take 
them.  Still  others  would  have  us  believe  that  the  public  is  de¬ 
manding  such  changes  as  these,  and  that  it  is  only  a  matter  of 
time  until  algebra  and  geometry  as  cultural  studies  are  relegated 
to  the  scrap  heap.  It  Is  a  matter  of  considerable  interest  to  know 
what  persons  outside  the  mathematical  world,  and  who  are 
qualified  to  speak  of  the  subject,  have  to  say  as  to  the  impor¬ 
tance  of  mathematics  in  our  scheme  of  education. 

Our  committee,  representing  the  Chicago  Mathematics  Club, 
sent  the  following  letter  to  prominent  doctors,  lawyers,  mer¬ 
chants,  bankers,  etc.,  in  the  city  of  Chicago.  We  have  avoided 
for  the  most  part  members  of  the  teaching  profession : 

“The  undersigned  committee  of  the  Chicago  Mathematics 
^Club,  in  cooperation  with  a  National  Committee  of  the  Mathe- 
L  matical  Association  of  America,  is  investigating  the  reasons 
for  teaching  mathematics  in  our  secondary  schools.  We  re- 
2  spectfully  submit  the  following  questions  to  you  in  order  that 
_  we  may  have  the  benefit  of  your  practical  experience.  Further¬ 
ed  more,  your  name  will  give  to  your  answers  a  weight  not  attached 
to  the  opinions  of  persons  less  well  known: 

“1.  Was  the  study  of  high  school  mathematics  worth  while  to  you?  If  so,  why? 

“2.  Did  its  study  contribute  to  your  development  anything  which  could  not  have  been 
__  secured  in  equal  degree  from  some  other  study  which  might  have  been  substituted  for  it? 

“3.  In  employing  a  young  person  or  in  advising  him  to  follow  your  profession,  what  impor- 
*  tance  would  you  attach  to  a  good  school  record  in  mathematics? 

^  ‘  ‘4.  In  planning  the  secondary  education  of  your  son  or  daughter,  would  you  include  algebra 

—and  geometry?  Why? 

“5.  Is  a  knowledge  of  algebra  or  geometry  of  any  practical  value  in  your  business?  If  so, 

-  in  what  way? 

(3—  “6.  Do  you  think  algebra  and  geometry  should  be  retained  in  our  secondary  schools? 

“Your  answers  to  these  questions  will  be  of  great  value  to 
those  who  are  striving  to  secure  for  our  schools  the  best  possible 

'This  part  of  the  report  was  read  before  the  Club  at  its  regular  October  meeting,  October 
6,  1917. 


26 


SCHOOL  SCIENCE  AND  MATHEMATICS 


course  of  study,  and  your  assistance  will  be  greatly  appreciated 


by  our  committee.” 

Fifty-five  replies  have  been  received  as  follows: 


Physicians  and  surgeons . .  7 

Lawyers . 12 

Bankers . .  7 

Merchants .  4 

Engineers.. .  1 

Clergymen. . .. .  9 

Manufacturers . .  2 

Authors  and  newspaper  writers— .  2 

Social  workers . .. .  1 

Business  managers . .  1 

Economist— . .  1 

Real  estate  men . .  1 

Broker _ _ _ 1 

Typefounder _ : . 1 

Railroad  men— . 2 

Secretaries .  2 

Printers . . 1 

Answers  to  question  1 : 

Affirmative. . .  43 

Very  emphatic— . 7 

Reasons  assigned: 

Mental  training . . . .  27* 

Training  in  accuracy....— .  5 

Training  in  concentration . .  3 

Pleasure.. .  3 

Practical  value _ _ 2 

Preparation  for  engineering— .  2 

Preparation  for  physicians . .  3 

Preparation  for  college.. .  3 

Negative . . 1 .  6 

Reasons: 

Never  used  it. . 1 

Of  little  value. . .  1 

Answers  to  question  2 : 

Affirmative . .  36 

Very  emphatic _  9 

Uncertain— .  8 

Negative. . . .  6 

(No  reasons  given,  no  comment.) 

Answers  to  question  3 : 

Great  importance. . 19 

Banker .  3 

Engineers .  1 

Physicians . 2 

Lawyers .  3 

Merchants .  2 

Clergymen. .  2 

Brokers . . 1 

Printers . .  1 

Manufacturers . .  2 

Railroad  men _  1 

Considerable  importance .  21 

Bankers. _  3 

Physicians . 2 

Economists _  1 

Lawyers .  7 


Clergymen. .  4 

Railroad  men . . 1 

Real  estate  men . . 1 

Secretaries . 1 

Very  little  importance. . .  2 

Lawyer  and  social  worker. 

No  importance . . 3 

Physicians _ _ _ _ .... _ _ _  2 

Typefounder. . 1 

Answers  to  question  4 : 

Both . ! _  4 2 

Algebra  at  least .  3 

Neither . .  2 

Only  as  a  prerequisite  for  professonal 
work .  3 


Answers  to  question  5 : 

Affirmative _ ’. _ _ _ 23 

Physicians . . . . . . . .  3 

1,  Biological  chemistry,  microscopic 
study. 

1,  Absolutely  essential  for  science 
involved  in  medicine. 

1,  Necessary  in  vital  statistics. 

Lawyers . 5 

Engineers— .  1 

Bankers . . 4 

1,  Investment  mathematics. 

1,  Refers  to  the  graph  as  important. 

Writers— . 1 

Mentions  its  value  in  allusion. 

Ministers . 2 

1,  Logical  help. 

1,  Argument  and  persuasion. 

Real  estate  man . . 1 

Manufacturers . . . .  2 

Economists . . 1 

Statistics  and  city  planning. 

Printer. _ _ _ 1 

Railroad  man . .  1 

Negative . .  26 

Bankers . . 3 

Lawyers .  5 

Clergymen. . . —  5 

Merchants .  3 

Physicians . 4 

Social  worker 
Broker 
Typefounder 
Railroad  man. 


Answers  to  question  6: 


Affirmative. . . 46 

Very  emphatic . .  14 

For  professional  preparation  only—. .  2 

Optional— .  2 

Negative . .  2 


The  following  additional  information  contained  in  the  replies 


is  of  considerable  interest: 


Mr.  John  G.  Shedd,  President,  Marshall  Field  and  Com¬ 
pany:  “I  firmly  believe  in  the  value  of  mental  training  derived 
from  the  study  of  mathematics,  both  the  elementary  and  the 
higher  branches.  I  know  that  training  has  been  of  especially 
great  value  to  me,  and  the  principles  are  constantly  used  in 
everyday  business  transactions.  It  seems  to  me  that  the  aim  of  all 
branches  of  a  school  curriculum  should  be  the  development 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


27 


of  analytical  and  logical  thinking  by  the  student.  Nothing 
in  my  judgment  tends  more  to  this  than  thorough  training  in 
mental  and  written  mathematics.” 

Mr.  C.  F.  Hoerr,  President,  Home  Bank  and  Trust  Company : 
“I  should  certainly  consider  it  a  serious  mistake  if  algebra  and 
geometry  were  not  a  part  of  the  education  of  our  children.  I 
should  want  them  to  have  algebra  and  geometry  for  a  number 
of  reasons: 

“1.  For  the  development  of  their  reasoning  and  logical  powers,  and  the  concentration  these 
subjects  will  demand. 

"2.  In  case  the  children  should  want  to  major  in  any  of  the  sciences. 

“3.  As  a  matter  of  general  culture  and  training  whether  they  desire  to  so  major  or  not. 

“Statistics  are  coming  into  their  own,  and  the  question  of 
graphs  is  very  important.  We  are  making  use  of  graphs  in  our 
business,  and  I  feel  that  through  this  medium  of  presenting 
statistics  we  are  better  posted  in  our  business.” 

Mr.  G.  M.  Reynolds,  President  of  the  Continental  and  Com¬ 
mercial  National  Bankj  “In  planning  the  education  of  a  child 
of  my  own  I  should  insist  upon  algebra  and  geometry;  for  while 
only  a  few  cases  might  occur  in  which  the  business  man  would 
make  practical  use  of  either,  the  mental  training  acquired  in 
gaining  a  knowledge  of  algebra  and  geometry  is  of  very  great 
help  in  directing  modern  business,  which  is  constantly  becoming 
more  complex  by  reason  of  increasing  burdens — a  condition 
which  calls  for  the  highest  degree  of  preparation  that  the  schools 
can  give.” 

Mr.  James  B.  Forgan,  Ex-President,  First  National  Bank, 
answers  the  questions  as  follows: 

“1.  Fifty  years  have  passed  since  I  studied  mathematics  in 
school.  I  have  always  thought  that  it,  better  than  any  other 
study  developed  my  mind  along  the  line  of  concentration  of 
thought  on  problems  confronting  me.  It'  was  a  study  I  liked 
and  in  which  I  excelled. 

“2.  Yes,  the  power  of  concentration,  and  I  know  of  no  other 
subject  that  if  substituted  for  it,  in  my  case,  could  have  at¬ 
tained  as  good  results. 

“3.  I  would  regard  a  good  school  record  in  mathematics  as 
indicating  a  mind  trained  to  concentration  of  thought  on  prob¬ 
lems  until  they  are  solved. 

“4.  Yes. 

“5.  Arithmetic  sufficient. 

“6.  I  do.” 

Rabbi  Emil  G.  Hirsch  answers  as  follows: 

“1.  It  was.  I  had,  even  as  a  lad,  a  knack  for  languages.  Be- 


28 


SCHOOL  SCIENCE  AND  MATHEMATICS 


fore  I  was  twelve  years  of  age  I  spoke  four  and  read  eight. 
Mathematics  provided  for  me  the  balance  wheel. 

“2.  It  did.  It  taught  me  to  reason  in  abstract  relations. 

“3.  Soundness  of  judgment,  I  hold, is  apt  to  be  better  devel¬ 
oped,  and  sequence  of  thought  as  well. 

“4.  I  should  include  both.  Educationally  both  are  effective. 
Practical  advantages — perhaps  the  discovery  of  mathematical 
bent  in  him. 

“5.  Not  directly  but  indirectly  it  is — reasons  given  above. 

“6.  I  do  most  emphatically.  High  schools  are  not  factories 
turning  out,  one  tracked  minds.” 

Mr.  H.  H.  Kennedy,  lawyer,  Moses,  Rosenthal  and  Kennedy, 
answers  as  follows: 

“1.  Unqualifiedly  yes.  It  is  my  judgment  that  the  study  of 
mathematics  very  materially  assists  in  developing  the  reasoning 
faculties  of  the  mind,  and  powers  of  anaylsis. 

“2.  Yes,  because  I  do  not  believe  there  is  any  study  taught 
in  the  high  school  that  would  tend  to  develop  in  an  equal  degree 
the  faculties  of  the  mind  referred  to.  Logic,  in  my  judgment, 
is  merely  a  higher  form  of  mathematics;  a  person  who  is  a  good 
mathematician  is  invariably  a  good  logician. 

“3.  In  advising  a  young  man  to  enter  upon  the  profession 
of  the  law,  I  would  urge  him  to  take  up  mathematics  above  all 
other  subjects  in  the  high  school,  since  I  believe  that  the  primary 
essential  to  qualify  a  person  to  successfully  practice  law  is  the 
mental  faculty  of  reasoning  and  the  power  of  analysis.  It  also 
tends  to  develop  the  powers  of  abstract  thought  to  a  much 
higher  degree  than  any  other  subject  taught  in  the  high  school 
and  especially  the  powers  of  concentration  essential  to  success 
in  any  profession,  and  more  particularly  the  law.  Should  a 
young  man  come  to  me  with  an  exceptionally  good  record  in 
mathemat  cs,  I  should  conclude  therefrom  that  he  had  a  well 
developed  mind,  and  was  especially  fitted  to  practice  law,  for 
the  ability  to  reason  well  and  the  power  to  concentrate  are  evi¬ 
dence  of  a  good  legal  mind. 

“4.  Yes,  I  would  most  certainly  include  these  two  studies  in 
planning  for  the  secondary  education  of  my  son;  indeed,  I 
pursued  this  very  course,  and  in  connection  with  his  studies 
at  the  University  High  School  and  Princeton  University  I  urged 
upon  him  the  importance  of  these  subjects,  and  he  pursued 
this  course,  his  intention  being  to  enter  later  upon  a  legal 
education. 

“5.  A  lawyer  enjoys  a  particular  advantage  if  he  is  well 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


29 


equipped  in  algebra  and  geometry;  he  finds  himself  brought 
into  all  forms  of  intellectual  inquiry  which  require  abstract 
thinking;  familiarity  with  these  subjects  comes  into  play  every 
single  day;  and  the  principles  of  geometry  are  the  foundation 
of  logic. 

“6.  By  all  means.  The  science  of  algebra  deals  with  repre¬ 
sentations  of  facts  and  permits  the  easier  handling  of  them  by 
the  mind.  A  person  well  versed  in  algebra  is  especially  qualified 
to  detach  himself  from  the  facts  of  life  in  such  a  way  as  to  enable 
him  to  reach  conclusions  free  from  bias  or  personal  interest. 
It  is  unquestionably  true  that  a  lawyer  who  is  a  good  mathema¬ 
tician  is  almost  invariably  a  lawyer  of  ability,  because  he  has  a 
trained  mind  that  enables  him  to  stand  upon  his  feet  and  reason 
well.” 

Thomas  E.  Donnelley,  President,  R.  R.  Donnelley  and  Sons 
Company,  Director  of  the  Chicago  Directory  Company,  answers 
as  follows: 

“1.  Yes,  it  trained  me  to  think  along  logical  lines  and  it 
gave  me  a  clear  idea  of  the  necessity  of  be  ng  100  per  cent 
accurate. 

“2.  Yes,  there  is  no  study  that  could  possibly  take  its  place. 

“3.  I  would  not  employ  him  unless  he  was  good  at  mathe¬ 
matics. 

“4.  Yes,  both,  and  trigonometry  as  well.  It  is  the  science  of 
facts  as  they  are.  Anyone  who  doesn’t  understand  these  studies 
lacks  an  appreciation  that  things  happen  because  they  must. 

“5.  Yes,  it  gives  one  the  ability  to  solve  a  problem. 

“6.  Yes.  Who  on  earth  thinks  otherwise?  It  is  the  basis 
of  our  whole  industrial  life.” 

Judge  E.  0.  Brown  of  the  Appellate  Court  answers  as  follows: 

“1.  It  was.  I  cannot  for  myself  conceive  why  mathematical 
truths  (which  seem  to  me  to  be  the  thoughts  of  God  before  time 
was)  should  not  be  worth  any  man’s  study. 

“2.  It  did. 

“3.  I  should  advise  any  young  person  to  study  mathematics 
and  to  obtain  as  good  a  school  record  as  he  could  in  them. 

“4.  I  would — and  did  with  my  three  sons  and  two  daughters. 
Because  I  consider  them  very  important  branches  of  knowledge 
and  very  stimulating  in  intellectual  pursuits. 

“5.  I  hardly  know  what  meaning  to  attach  to  ‘practical.’ 
I  should  be  less  competent  in  every  way  without  such  knowledge. 

“6.  I  certainly  do.  If  they  are  to  be  eliminated  I  do  not  know 
what  should  be  retained.” 


30 


SCHOOL  SCIENCE  AND  MATHEMATICS 


Dr.  James  B.  Herrick,  physician:  “ A  progressive  physician 
today  must  know  physics,  chemistry,  physiology,  physical 
chemistry,  etc.  Without  the  elements  of  mathematics  such 
knowledge  is  impossible.” 

Mr.  Marquis  Eaton,  attorney:  “These  studies  (algebra  and 
geometry)  tend,  in  my  opinion,  to  develop  the  power  of  analysis, 
which  power  lies  at  the  threshold  of  success  in  every  profession, 
notably  the  legal  profession.” 

Mr.  James  E.  Otis,  President,  Western  Trust  and  Savings 
Bank,  “I  consider  the  study  of  high  school  mathematics  as  a 
necessary  part  of  a  high  school  education.  In  so  far  as  I  am 
concerned,  my  knowledge  of  algebra,  geometry  and  higher 
mathematics  frequently  enables  me  to  analyze  problems  which 
come  before  me.  It  is  the  training  of  the  young  mind  in  the 
study  of  algebra  which  I  consider  of  greatest  importance  and 
not  the  practical  use  he  can  make  of  the  science  in  later  life. 
.In  so  far  as  geometry  is  concerned,  questions  are  constantly 
arising  where  I  find  my  knowledge  of  geometry  most  helpful. 
I  feel  so  strongly  on  the  subject  that  I  should  consider  the  drop¬ 
ping  of  algebra  and  gometry  in  our  secondary  schools  as  a  very 
serious  mistake” 

Mr.  S.  M.  Hastings,  President,  Computing  Scales  Com¬ 
pany  of  America:  “My  personal  business  experience  has 
taught  me  that  mathematics,  algebra  and  geometry  are  impor¬ 
tant,  and  in  my  opinion  they  should  be  retained  in  our  schools. 
With  a  knowledge  of  these,  one  is  better  fitted  to  cope  with  the 
business  problems. 

“As  we  become  a  greater  world  power  and  our  foreign  business 
expands,  the  need  of  every  possible  mental  equipment  will  be 
more  and  more  in  evidence.” 

Mr.  C.  A.  Creider,  Secretary  to  Edward  B.  Butler  of  Butler 
Brothers:  “Anyone  who  has  been  able  to  compass  the  subjects 
of  algebra  and  geometry  is  by  very  nature  of  his  training  equipped 
to  comprehend  a  problem  or  to  arrive  at  a  solution  thereof 
much  more  readily  than  the  fellow  who  has  not  had  the  training 
referred  to.  In  other  words,  even  a  private  secretary,  who  might 
be  expected  to  have  very  little  to  do  with  the  principles  of  these 
two  studies,  finds  it  much  easier  to  fill  his  position  having  gone 
through  the  mental  exercise  of  study  of  these  two  subjects.” 

Mr.  C.  S.  Cutting,  attorney,  Holt,  Cutting  and  Sidley:  “The 
study  of  high  school  mathematics  was  worth  while  to  me  for 
this  reason,  among  others,  that  the  cultivation  of  the  habit  of 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


31 


mathematical  methods  of  thought  not  only  fixed  my  knowledge 
of  the  so-called  practical  mathematics  of  the  grammar  school, 
but  made  all  operations  of  like  nature,  whether  strictly  mathe¬ 
matical  or  otherwise,  easier.  .  .  It  is  not  usually  possible  to 
shirk  work  (as  a  student)  in  mathematics  (it  is  therefore  de¬ 
sirable  in  the  course  of  study).  .  .  .  We  use  geometry  in  the 
examination  of  titles  to  real  estate,  in  the  investigation  of 
plats  and  in  many  other  relations  where  mensuration  of  both 
lines  and  angles  must  be  considered.  In  my  opinion,  however, 
this  is  the  smallest  part  o  the  benefit  derived  from  the  study 
of  these  subjects.  ...  I  believe  that  the  emasculation  of  a 
literary  college  course  by  omitting  the  classics  is  an  educational 
mistake  which  would  be  equaled  by  omitting  the  mathematics. 
IVty  experience  teaches  me  that  the  man  who  knows  a  few 
things  well  is  better  educated  than  the  one  who  knows  many 
things  indifferently.” 

Mr.  James  K.  Ingalls,  President,  Western  Heater  Despatch: 
“High  school  mathematics  taught  me  system,  to  be  accurate, 
to  have  confidence  in  my  conclusions,  and  accustomed  me  to 
abstract  reasoning.” 

Mr.  Granger  Farwell  broker:  “I  employ  algebraic  and 
geometrical  calculations  in  business  and  many  times  without 
knowing  that  I  specifically  do  so.” 

Miss  Harriet  Vittum,  social  worker,  says: 

“2.  Yes,  I  disliked  it  so  heartily  I  needed  to  have  to  do  it. 

“6.  Yes,  as  general  mental  discipline  if  not  to  fill  specific 
needs.” 

Rev.  James  G.  K.  McClure  President,  McCormick  Theo¬ 
logical  Seminary,  “I  am  accustomed  to  say  that  the  classics 
have  done  more  to  prepare  me  for  my  life  than  any  other  study  or 
studies.  The  drill  in  Euclid  at  Yale  was  very  disciplinary.” 

Rev.  M.  P.  Boynton,  Woodlawn  Baptist  Church:  “The 
practical  geometry  that  I  learned  in  my  father’s  carpenter  shop, 
and  which  was  based  on  the  steel  square  of  the  workman,  has 
been  of  greatest  practical  use  to  me.  Though  I  had  no  occasion 
to  use  these  rules  for  many  years,  I  was  delighted  when  in  build¬ 
ing  my  summer  home,  and  needing  the  rules  for  cutting  and  fit¬ 
ting  the  rafters  and  other  timbers  to  find  that  these  principles 
wh  ch  I  actually  worked  out  as  a  boy  freely  came  back  to  me. 
I  believe  that  this  is  nature’s  great  law  and  that  we  retain  only 
such  knowledge  as  we  are  able  to  make  real  use  of.” 

Dr.  J.  Clarence  Webster,  physician,  says  that  in  planning 


32 


SCHOOL  SCIENCE  AND  MATHEMATICS 


the  education  of  a  son  or  daughter  he  would  not  include  algebra 
or  geometry,  “unless  these  subjects  were  directly  related  to 
their  professional  work.  ...  I  think  they  are  as  useless  as  Latin 
and  Greek  and  should  not  be  taught  to  the  great  majority.” 

A  prominent  clergyman  of  the  city  writes:  “Even  the  ad¬ 
mitted  value  of  geometry  in  training  the  logical  faculties  has  to 
be  discounted  because  the  faculty  is  there  developed  in  a  field 
entirely  apart  from  any  of  the  practical  problems  of  life,  and  its 
general  cultural  value  is  therefore  indirect  and  remote.  This  is 
not  so  much  a  matter  of  judgment  on  my  part  as  a  matter  of 
practical  experience.  My  judgment  is  that  algebra  and  geometry 
are  quite  unnecessary  as  a  part  of  general  education  and  that 
for  purposes  of  general  culture  other  subjects  are  much  more 
useful  in  the  curriculum  of  secondary  education.  Yet  it  is  per¬ 
fectly  clear  that  engineering  and  architecture,  not  to  mention 
certain  branches  of  physics  and  astronomy,  absolutely  require 
_a  knowledge  of  algebra  and  geometry. 

“My  daughter,  now  twenty-one  years  of  age,  had  a  distinct 
gift  for  languages  and  history.  Mathematics  of  any  kind  were 
exceedingly  difficult  for  her  and  even  untiring  industry  and  ex¬ 
cessive  efforts  barely  sufficed  to  secure  a  passing  mark.  Yet 
algebra  was  a  required  study  and  graduation  from  high  school 
impossible  without  a  reasonably  satisfactory  ^record  in  this 
study.  She  will  never  make  any  use  of  algebra  and  she  has 
bitter  memories  connected  with  its  study,  together  with  a  rank¬ 
ling  sense  of  injustice  in  a  system  which  requires  a  given  standard 
in  this  useless  subject  and  allows  no  substitutes  in  the  require¬ 
ments  for  graduation.” 

This  is  probably  as  representative  a  group  of  persons  as 
could  be  selected.  A  larger  group  would  not  be  likely  to  change 
results  greatly.  They  are  all  busy  men  of  affairs.  That  they 
should  answer  so  fully  is  significant  of  the  importance  they 
attach  to  such  matters.  About  90  per  cent  answer  favorably 
to  the  importance  6f  mathematics  in  secondary  schools,  some  of 
them  emphatically  so.  This  is  an  overwhelming  majority. 
An  election  with  such  a  majority  would  be  called  “a  landslide.” 
While  there  are  many  factors  which  may  enter  to  make  the 
opinion  of  a  single  individual,  or  even  of  a  small  minority,  of 
doubtful  value,  great  importance  must  be  attached  to  the 
opinion  of  such  a  majority.  The  members  of  such  a  group 
cannot  all  be  biased  or  prejudiced;  they  cannot  all  be  out  of  date 
in  such  matters,  or  be  unacquainted  with  the  trend  of  affairs  in 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


33 


the  educational  world.  The  fact  that  they  are  successful  men 
precludes  any  such  notion.  Some  of  them  are  giants  in  the  busi¬ 
ness  world  and  in  other  fields — men  to  whom  we  look  for  leader¬ 
ship. 

However,  the  opinion  of  a  minority,  or  even  of  a  single  in¬ 
dividual,  is  not  a  negligible  matter.  These  frequently  suggest 
important  lines  for  improvement,  or  they  raise  an  issue  and  put 
to  the  test  things  that  have  come  to  us  through  tradition.  When 
we  recognize  a  fallacy  in  the  argument  of  an  opponent  we  are 
reassured  in  the  position  we  hold.  Those  who  have  opposed  the 
teaching  of  mathematics  in  high  schools — where  they  have  ex¬ 
pressed  themselves  beyond  the  mere  answer  “no” — seem  to 
think  of  mathematics  only  as  a  tool.  To  them  its  only  value 
seems  to  be  the  direct  application  in  the  sciences  or  in  the 
doing  of  the  world’s  work.  This  is  a  mistake,  as  is  shown  by 
the  answers  to  questions  one  and  five.  Question  one,  which  is 
considered  as  referring  to  cultural  values,  has  an  overwhelming 
majority  in  the  affirmative;  while  question  five,  which  repre¬ 
sents  the  utility  values,  has  a  small  negative  majority.  Evidently 
the  direct  application  of  mathematics  to  other  fields  is  of  secon¬ 
dary  importance.  The  chief  values  are  culture  and  mental  dis¬ 
cipline,  training  in  logical  and  abstract  thinking,  the  cultiva¬ 
tion  of  the  power  of  concentration,  and  the  acquiring  of  speed 
and  accuracy  in  our  mental  activities.  However,  this  matter 
will  be  treated  at  greater  length  in  the  more  complete  report  to 
be  made  later  by  this  committee. 

President  Butler  of  Columbia  University  says  in  Educational 
Review,  September,  1917:  “No  educational  instruments  have 
yet  been  found  that,  in  disciplinary  value,  are  equal  to  Greek, 
Latin,  and  mathematics.  The  descriptive  and  experimental 
sciences  cannot  do  it — or  at  least  they  have  not  done  it — and 
the  same  is  true  of  the  newer  subjects  of  study  that  are  humor¬ 
ously  if  roughly,  classified  together  as  ‘unnatural  sciences’ — 
economics,  sociology,  and  the  like.” 

An  investigation,  as  reported  by  Mr.  Harris  Hancock  in 
School  and  Society,  June  19,  1915,  was  carried  out  recently 
in  Cincinnati.  Of  105  prominent  persons  in  the  city,  and  99 
outside  the  city,  consulted  with  reference  to  the  question, 
“What  course  of  study  should  be  taken  by  a  boy  who  is  enter¬ 
ing  high  school?”  167  answered  that  they  would  require  mathe¬ 
matics,  18  would  require  either  mathematics  or  the  classics,  and 
12  would  make  mathematics  elective.  This  experiment  carried 


34 


SCHOOL  SCIENCE  AND  MATHEMATICS 


on  with  a  larger  group  more  widely  scattered  gives  approxi¬ 
mately  the  same  majority  in  favor  of  mathematics  as  is  shown 
in  our  experiment.  The  following  came  with  the  replies: 

Hon.  Chas.  Theodore  Greve,  of  the  Cincinnati  Law  School: 
“I  have  specialized  in  history  and  economics,  but  they  can  never 
take  the  place  of  classics  or  mathematics.  Both  are  essential 
and  there  can  be  no  possible  substitute  for  either.  .  .  .  These 
two  subjects  are  the  essential  groundwork  of  education  that  is 
to  be  of  value  for  any  subsequent  career,  professional,  scientific, 
business  or  mechanical.” 

Hon.  Roscoe  Pound,  of  the  Harvard  Law  School:  “The 
two  things  which  appear  to  me  to  be  required  of  secondary 
education  are,  linguistic  training  .  .  .  and  some  sort  of  train¬ 
ing  which  will  form  settled  mental  habits  of  accuracy  and  thor¬ 
oughness  during  the  student’s  formative  period.  I  believe 
mathematics  will  achieve  the  latter  result  as  nothing 
else  may.  Personally  I  never  liked  mathematics;  but  they  were 
valuable,  not  for  what  I  remembered,  but  because  I  was  com¬ 
pelled  to  see  that  two  and  two  make  four,  instead  of  presenting 
plausible  arguments  that  it  might  be  otherwise.” 

Mathematics  is,  however,  an  important  study  not  for  the  boy 
alone  but  for  the  girl  as  well.  Since  the  girls  of  today  will  be 
the  mothers  of  tomorrow,  and  so  will  bear  the  greater  share  of 
responsibility  in  the  education  of  the  coming  generation,  for 
this  reason,  if  for  no  other,  algebra  and  geometry  ought  to  be  a 
part  of  the  education  of  every  girl. 

Margaretta  Tuttle,  an  author,  and  also  of  Cincinnati,  says 
in  Good  Housekeeping,  September,  1917,  that  the  old  line  edu¬ 
cation,  even  though  requiring  effort  that  is  not  always  pleasant, 
is  best.  “Will-power  continues  to  be  the  great  need  of  all  who 
hope  to  grow  and  be  of  use.  And  it  continues  to  develop  only 
by  use. 

“Marianne  hates  solid  geometry.  Marianne’s  mother  does 
not  see  why  her  daughter  has  to  waste  a  whole  year  on  some¬ 
thing  she  can  never  by  any  possibility  use.  Marianne  is  liter¬ 
ary.  Oh,  no!  Literary  ability  and  mathematical  ability  rarely 
go  together.  Marianne,  who  is  eighteen,  and  who  knows  a  little 
about  the  lives  of  writers  and  nothing  at  all  about  engineers, 
will  tell  you  so.  But  all  the  same,  after  Marianne,  by  the 
sweat  of  her  brow,  has  worked  out  these  different  problems, 
she  will  have  formed  the  habit  of  not  flinching  from  thinking 
things  out — one  of  the  most  necessary  habits  for  a  literary 
person  and  for  any  woman. 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


35 


“Women  do  not  always  do  it,  nor  men  either.  It  is  the  think¬ 
ing  of  things  out  that  steadies  the  world.  It  is  the  confronting 
of  a  problem  with  the  intent  of  solving  it  after  you  have  seen 
both  sides  of  it  that  makes  you  a  real  help  in  a  world  of  unsolved 
problems.  You  cannot  write  a  good  story,  short  or  long,  with¬ 
out  a  problem  in  it  that  has  to  be  thought  out  and  then  demon¬ 
strated.  Many  a  literary  person  living  a  meager  life  on  small 
success,  unable  to  solve  the  problem  of  why  it  is  so  meager,  needs 
a  little  mathematical  training.  Many  a  mother  struggling 
with  the  problem  of  her  child’s  food,  and  wishing  she  had  taken 
dietetics  instead  of  piano  lessons,  needs  the  mathematical  touch 
— the  ability  first  to  recognize  a  problem  when  she  sees  it,  then 
to  attack  it  without  worry,  to  bring  to  its  solution  all  the  like 
problems  that  have  been  previously  solved,  all  the  axioms 
and  theorems  that  actually  apply  to  it,  and  then  to  come  to  a 
conclusion  and  to  stick  to  it  because  it  has  been  demonstrated. 
Quod  erat  demonstrandum !  The  woman  who  has  learned  the 
full  power  of  that  phrase  has  learned  to  handle  facts.” 

These  testimonies  and  the  results  of  these  investigations 
make  it  evident  that  algebra  and  geometry  are  not  so  much  rub¬ 
bish  to  be  thrown  out  at  the  back  door  as  quickly  as  possible, 
as  some  would  have  us  think.  The  number  of  persons  who  ob¬ 
ject  to  the  requiring  of  mathematics  in  our  high  schools  is  rela¬ 
tively  a  small  minority,  which,  in  its  efforts  to  be  heard,  some¬ 
times  makes  a  noise  all  out  of  proportion  to  its  numbers.  The 
professional,  the  business,  and  the  industrial  worlds,  at  least, 
demand  for  the  present,  and  will  demand  for  years  to  come,  that 
algebra  and  geometry  be  retained  as  required  studies  in  the 
secondary  schools  of  our  country;  and  this  notwithstanding 
the  efforts  of  educational  theorists  and  would-be  reformers 
to  the  contrary.  Would  it  not  be  time  and  effort  better  spent 
if  the  destructionists  were  to  aim  at  the  perfecting  of  the  sub¬ 
ject  matter,  and  to  the  improvement  of  its  teaching? 

The  Committee: 


ALFRED  DAVIS, 

Chairman, 

Francis  W.  Parker  School, 
330  Webster  Ave.,  Chicago,  Ill. 
A.  M.  ALLISON, 

Lake  View  High  School, 

Chicago,  Ill. 

J.  A.  FOBERG, 

Crane  Technical  High  School, 

Chicago,  Ill. 


M.  J.  NEWELL, 

Evanston  High  School, 
Evanston,  Ill. 
C.  M.  AUSTIN, 

Oak  Park  High  School, 
Oak  Park,  Ill. 
J.  R.  CLARK, 

President  (Ex-Officio) 
Parker  High  School,  Chicago,  Ill. 


112 


SCHOOL  SCIENCE  AND  MATHEMATICS 


VALID  AIMS  AND  PURPOSES  FOR  THE  STUDY  OF 
MATHEMATICS  IN  SECONDARY  SCHOOLS. 

By  Alfred  Davis, 

Francis  W.  Parker  School, 

330  Webster  Ave.,  Chicago. 

Chairman  of  a  Committee  of  the  Mathematics  Club  of  Chicago 
Appointed  to  Investigate  This  Topic. 

It  is  our  purpose  in  making  this  report  to  aid  the  National 
Committee  of  the  Mathematical  Association  of  America  in 
investigating  the  reasons  for  the  teaching  of  mathematics  in 
our  secondary  schools.  The  topic  is  a  vital  one,  since  its  con¬ 
sideration  must  determine  the  place  and  the  nature  of  the 
mathematics  taught  in  our  high  schools.  Its  discussion  is 
timely,  since  the  challenge  has  come  from  various  sources  to 
us,  as  teachers  of  mathematics,  to  defend  our  subjects,  espe¬ 
cially  algebra  and  geometry.  Educational  inertia  will  no  longer 
protect  us.  A  passive  attitude  is  no  longer  tenable;  we  must 
make  our  position  clear.  Even  though  we  have  little  that  is 
new  by  way  of  argument  to  offer;  and  even  though  we  prove 
nothing  beyond  the  possibility  of  a  question,  a  statement  of 
what  we  are  convinced  is  true  cannot  fail  to  be  of  value  both 
to  teachers  and  to  laymen  whom  it  may  reach.  Since  it  is 
desirable,  if  possible,  to  avoid  every  generalization  not  supported 
by  the  results  of  scientific  investigation  or  by  the  highest  philo¬ 
sophic  authority,  we  shall,  throughout  the  discussion,  make 
generous  use  of  the  investigations  of  others  and  shall  quote 
freely  from  those  who  can  speak  with  authority.  We  make 
bold  to  do  this  because  important  statements  and  valuable 
work  increase  in  importance  with  emphasis  and  use.  In  the 
words  of  Dr.  Smith,  of  Teachers  College,  when  speaking  of 
philosophers  of  standing  and  of  those  who  are  masters  in  their 
respective  fields  of  effort,  “These  after  all  are  the  men  who  are 
our  leaders.” 

Part  I. 

The  study  and  the  teaching  of  mathematics  in  our  secondary 
schools  may  be  justified  under  the  following  heads,  and  it  is  to 
these  that  we  may  turn  to  establish  the  validity  of  our  aims  and 
purposes : 

1.  Philosophy. 

2.  Psychology  and  Its  Experiments. 

3.  Experience. 

4.  The  Utility  oj  Mathematics 


/ 


MATHEMATICS  IN  SECONDARY  SCHOOLS  113 

1.  Philosophers  from  Plato  to  the  present  time  have  been 
almost  unanimous  in  their  approval  of  mathematical  studies. 
They  believe  that  culture  and  mental  discipline  result  from  their 
pursuit.  These  are  men  to  whom  we  must  listen.  They  are 
masters  in  the  art  of  reasoning  and  arrive  at  conclusions  free 
from  sentiment  or  prejudice.  Where  exceptions  occur,  it  is 
usually  due  to  a  lack  of  knowledge  of  mathematics  and  a  con¬ 
sequent  inability  to  judge.  The  famous  criticism  of  Sir  William 
Hamilton  has  no  weight,  since  John  Stuart  Mill  (“Examination 
of  Sir  Wm.  Hamilton’s  Philosophy,”  p.  607)  shows  that  Hamil¬ 
ton  did  not  know  mathematics;  and  Professor  C.  J.  Keyser 
(in  an  address  at  Columbia  University,  October  16,  1907) 
shows  that  he  was  prompted  by  unworthy  motives.  Oliver 
Wendell  Holmes  and  Thomas  Huxley  were  sincere  in  their 
doubts  as  to  the  value  of  the  study  of  mathematics,  but  these 
have  been  ably  answered  by  J.  J.  Sylvester  and  C.  J.  Keyser. 
Professor  A.  N.  Whitehead  (“Introduction  to  Mathematics,” 
p.  113)  says,  “Philosophers,  when  they  have  possessed  a  thor¬ 
ough  knowledge  of  mathematics,  have  been  among  those  who 
have  enriched  the  science  with  some  of  its  best  ideas.  On  the 
other  hand  it  must  be  said  that,  with  hardly  an  exception, 
all  the  remarks  on  mathematics  made  by  those  philosophers 
who  have  possessed  a  slight  or  hasty  and  late-acquired  knowl¬ 
edge  of  it  are  entirely  worthless,  being  either  trivial  or  wrong.” 
Philosophy’s  support  of  mathematics  stands  like  the  rock  of 
Gibralter,  secure  amid  the  passing  assaults  made  against  it. 

Plato,  “Let  no  one  ignorant  of  geometry  enter  my  door.” 

August  Comte,  “No  irrational  exaggeration  of  the  claims  of 
mathematics  can  ever  deprive  that  part  of  philosophy  of  the 
property  of  being  the  natural  basis  of  all  logical  education, 
through  its  simplicity,  abstractness,  generality,  and  freedom 
from  disturbance  by  human  passion.  There,  and  there  alone, 
we  find  in  full  development  the  art  of  reasoning,  all  the  resources 
of  which,  from  the  most  spontaneous  to  the  most  sublime,  are 
continually  applied  with  far  more  variety  and  fruitfulness  than 
elsewhere.  .  '  .  .  The  more  abstract  portions  of  mathematics 

may  in  fact  be  regarded  as  an  immense  repository  of  logical 
resources,  ready  for  use  in  scientific  deduction  and  coordination.” 

G.  Stanley  Hall,  “Mathematics  is  the  ideal  and  norm  of  all 
careful  thinking.” 

2.  It  was  but  yesterday  that  psychologists  denied  the  trans¬ 
fer  of  training,  and  so  to  them  the  ability  acquired  in  the  study 


114 


SCHOOL  SCIENCE  AND  MATHEMATICS 


of  mathematics  was  of  value  only  in  the  further  study  of  mathe¬ 
matics.  This  position  has  not  been  successfully  defended. 
Professor  Smith,  of  Columbia,  says,  “The  attack  upon  mathe¬ 
matics,  that  it  has  no  general  disciplinary  value,  has  thus  far 
been  abortive  scientifically.  We  have  only  to  note  how  di¬ 
vergent  are  the  results  of  various  investigations  to  see  the  truth 
of  the  assertion.’ ’  Professor  G.  M.  Stratton,  of  the  University 
of  California  (“School  and  Society”)  says;  “It  is  a  grave  mis¬ 
take  to  suppose  that  the  experimental  work  has  proved  that 
the  idea  of  mental  discipline  is  no  longer  tenable.”  Today  all 
psychologists  of  standing  concede  transfer.  The  only  questions 
are  as  to  how  much,  and  by  what  agencies  it  is  accomplished. 
Is  it  not  probable  that  tomorrow  all  doubts  regarding  the 
mental  transfer  of  training  will  have  run  their  course  and  will 
be  dead,  at  least  for  a  time? 

It  would  be  of  interest  to  show  that  questionable  results  have 
been  obtained,  and  that  undue  importance  has  been  attached 
to  experiments  worked  out  by  psychologists  and  others,  in  at¬ 
tempting  to  show  that  the  value  of  the  study  of  mathematics 
has  been  exaggerated.  Indeed,  when  we  consider  that  these 
experiments  are  likely  to  be  limited  in  scope;  that  the  material 
to  be  experimented  with  (human  minds)  is  almost  infinite  in 
possible  variety;  and  that  the  interpretation  of  results  is  largely 
an  individual  matter;  we  must  concede  that  such  efforts  are  of 
doubtful  value.  However,  psychology  and  its  experiments 
in  relation  to  the  teaching  of  mathematics  is  to  be  considered 
by  another  committee. 

3.  The  experience  of  the  race  justifies  the  teaching  of  mathe¬ 
matics.  This  is  almost  too  obvious  to  need  mention.  Civiliza¬ 
tion  may  be  measured  in  terms  of  mathematical  progress. 
Mathematics  is  the  ladder,  giving  the  sure  footing,  by  which 
we  have  climbed  steadily  to  higher  levels  of  achievment.  It  is 
through  mathematics  that  man  has  been  able  to  strip  mystery 
from  the  forces  in  nature  and  to  harness  them  for  his  service. 
Colonel  F.  W.  Parker  (“Talks  on  Pedagogics,”  p.  92)  says, 
“The  lower  the  grade  of  development  in  the  human  race  the  less 
there  is  known  of  number.”  Mathematics  is  then  an  intrinsic 
element  in  human-  progress.  Scientific  progress  is  impossible 
without  it.  It  is  interwoven  in  the  fabric  of  our  commercial  and 
industrial  life.  Since  this  is  true  it  must  change  as  the  race 
advances — perhaps  slowly — but  ever  to  increase  the  importance 
of  its  study. 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


115 


4.  The  direct  practical  utility  of  mathematics  alone  will  not 
justify  our  present  high  school  courses  in  the  subject.  To  many 
earnest  and  sincere  critics  mathematics  is  only  a  tool  invented 
by  man  for  the  mastery  of  other  fields.  In  fact,  this  is  the  rock 
on  which  most  of  those  who  object  to  mathematics  shipwreck. 
Mathematics  is  vastly  more  than  a  tool;  it  is  a  type  of  thought, 
and  even  high  school  students  gain  indispensable  training  in 
mental  activities  from  it — such  training  as  cannot  be  gained  so 
well  from  any  other  study.  Professor  C.  J.  Keyser  (“The  New 
Infinite  and  the  Old  Theology”)  says,  “Mathematics  is  indeed 
a  humble  servant — a  drudge,  if  you  please;  an  unsurpassed 
drudge — in  the  sense  that  nothing  else  does  a  larger  share  of 
humble  and  homely  work.  To  imagine,  however,  that  her  place 
in  the  hierarchy  of  knowledges  is  thereby  defined  is  hardly  the 
beginning  of  wisdom  in  the  matter.  It  is  necessary  to  look 
much  higher.  Her  rank  in  the  ascending  scale  is  not  that  of 
a  useful  drudge,  immeasurable  as  is  her  service  in  that  capacity; 
it  is  not  merely  the  rank  of  a  metric  and  computatory  art,  in¬ 
valuable  as  the  latter  is,  as  well  in  science  as  in  the  affairs  of  a 
workaday  world ;  it  is  not  even  that  of  a  servant  to  other  sciences 
in  their  fields  of  experimental  and  observational  research, 
indispensable  as  mathematics  is  in  that  regard;  over  and  above 
these  things,  she  is  charged  with  a  sacred  guardianship — in  her 
keeping  are  certain  ideals,  the  ideal  forms  of  science  and  the 
standards  of  perfect  thinking;  she  is  concerned  not  with  the 
vagaries,  but  with  the  verities  of  thought,  with  select  matters 
independent  of  opinion,  passion,  accident,  and  will;  it  is  thus 
peculiarly  hers  to  release  human  faculties  from  the  dominion 
of  sense  by  winning  allegiance  to  things  that  abide;  her  medita¬ 
tions  transcend  the  accidents  of  time  and  place;  it  is  their 
idiosyncrasy  to  have  for  subject  proper,  not  the  fickle  and  transi¬ 
tory  elements  in  the  stream  of  a  flowing  world,  but  those  aspects 
of  being  that  present  themselves  under  the  forms  of  the  infinita^— 
and  eternal.”^  The  tendency  today  is  to  overemphasize  the 
utility  of  all  high  school  studies.  Professor  John  Dewey  (“How 
We  Think,”  p.  138)  says,  “There  is  such  a  thing,  even  from  the 
commonsense  standpoint,  as  being  Too  practical/  as  being  so 
intent  upon  the  immediately  practical  as  not  to  see  beyond  the 
end  of  one’s  nose  or  as  cutting  off  the  limb  upon  which  one  is 
sitting.  .  .  .  Exclusive  preoccupation,  with  matters  of  use  and 
application,  so  narrows  the  horizon  as  in  the  long  run  to  defeat 
itself.  It  does  not  pay  to  tether  one’s  thoughts  to  the  post  of 


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use  with  too  short  a  rope.  Power  in  action  requires  some  large¬ 
ness  and  imaginativeness  of  vision.  Men  must  at  least  have 
enough  interest  in  thinking  to  escape  the  limits  of  routine  and 
custom.  Interest  in  knowledge  for  the  sake  of  knowledge,  in 
thinking  for  the  sake  of  the  free  play  of  thought,  is  necessary 
then  to  the  emancipation  of  practical  life — to  make  it  rich  and 
progressive.”  Lord  Beaconsfield’s  definition  of  a  practical 
man  is  too  often  the  truth:  “A  practical  man  is  one  who  prac¬ 
tices  the  errors  of  his  forefathers.” 

Many  today  make  the  mistake  of  measuring  educational, 
values  in  terms  of  money.  They  would  prepare  the  child  merely 
to  get  the  necessaries  of  life.  These  are  the  enemies  of  liberal 
education;  for  if  such  are  our  aims  we  are,  as  a  people,  in  the 
process  of  decay.  The  aim  of  education  should  be  to  make 
lives  worth  preserving — lives  that  will,  at  least  in  some  small 
measure,  make  the  world  a  better  place  in  which  to  live.  But 
multitudes  of  men  and  women  “like  dumb  driven  cattle”  go 
wearily  to  toil  each  day.  They  use  the  last  measure  of  strength 
in  earning  a  livelihood — beyond  this  they  have  no  vision.  If 
they  seem  satisfied  it  is  because  they  lack  the  outlook  on  life 
which  an  education  ought  to  give.  There  is  danger  that  edu¬ 
cators  will  assume  that  people  need  training  for  efficiency  in 
this  sort  of  life,  but  such  is  not  the  vision  of  leadership.  These 
people  need  the  best  possible  education,  as  much  of  it  as  they 
can  get;  not  an  education  in  any  sense  inferior,  or  suited  to  an 
inferior  station  in  life.  Professor  Keyser  ( Educational  Review, 
April,  1917)  says,  “I  desire  to  warn  you,  as  a  friend,  against 
the  enemies  of  liberal  education.  These  are  very  numerous, 
being  easy  to  produce,  springing  up  like  weeds  along  the  dusty 
highway,  almost  under  the  very  hoof  of  travel.  I  desire  to  warn 
you  against  the  insidious  and  baleful  influence  of  omnipresent, 
well-meaning,  wingless-minded  educators  who  unconsciously 
conceive  young  men  and  women  as  more  or  less  sublimated 
beasts;  and  who  regard  colleges  and  universities  as  agencies  for 
teaching  the  animals  the  arts  of  getting  shelter  and  raiment 
and  food.”  The  motto  of  the  Pythagorean  Brotherhood  should 

be  the  motto  of  today, 

“A  figure  and  a  step  onward: 

Not  a  figure  and  a  florin.” 

Yet  the  study  of  mathematics  will  ever  be  important  from  the 
standpoint  of  utility.  Geometry  originated  in  Egypt  from  the 
need  to  survey  the  farms  in  the  valley  of  the  Nile,  and  to  replace 
the  landmarks  swept  away  by  the  periodic  overflow  of  the  river. 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


117 


Any  schoolboy  knows  that  algebra  enables  us  to  solve  problems 
which  are  practically  impossible  of  solution  by  arithmetic.  The 
invention  of  the  calculus  enabled  Newton  to  apply  his  law  of 
gravitation  to  the  motions  of  the  planets.  As  needs  have  arisen 
in  various  fields,  mathematics  have  been  invented  to  relate 
theory  to  fact.  Discoveries  have  been  made  in  pure  mathe¬ 
matics,  when  studied  for  its  own  sake,  and  later  these  have  been 
applied  to  practical  ends.  There  are  many  fields  in  which  a 
knowledge  of  mathematics  is  absolutely  essential  to  their  mas¬ 
tery;  indeed,  it  seems  to  be  necessary  to  all  branches  of  knowl¬ 
edge  as  these  become  more  complete  and  so  more  scientific. 
Dr.  O.  J.  Lee,  of  Yerkes  Observatory,  calls  attention  to  the  fact 
that  many  astronomers,  while  in  the  opinion  of  the  outside 
world  successful,  have  failed  from  lack  of  a  sufficient  founda¬ 
tion  in  mathematics.  The  same  is  true  of  many  other  lines  of 
effort.  Professor  J.  W.  A.  Young  (“The  Teaching  of  Mathe¬ 
matics”)  says,  “For  direct  practical  usefulness,  mathematics 
is  second  only  to  the  mother  tongue.”  Regarding  the  possible 
future  of  mathematics  in  this  line,  G.  St.  L.  Carson  (“Mathe¬ 
matical  Education,”  p.  51)  says,  “I  believe  that  the  modern 
theories  of  pure  mathematics  are  destined  to  illumine  our 
understanding  of  the  human  mind  and  of  cities  and  nations, 
just  as  the  pure  mathematics  of  fifty  years  ago  has  already 
illumined  the  previously  dark  and  chaotic  field  of  physical 
science;  that  modern  mathematics  is  or  will  be  to  psychology, 
history,  sociology,  and  economics  as  has  been  the  older  mathe¬ 
matics  to  electricity,  heat,  light,  and  other  branches  of  physical 
science.” 

It  is  evident  that  the  study  of  mathematics  is  amply  justified 
by  the  testimony  of  philosophers  who  know;  by  its  importance 
as  an  aid  in  mental  development;  by  the  weight  of  human  ex¬ 
perience;  and  because  of  its  increasing  utility.  Mathematics 
is  an  essential  part  of  any  scheme  of  education  that  pretends 
to  be  well  balanced  or  complete.  Since  the  education  of  a  rapidly 
increasing  number  of  people  ends  with  the  high  school,  it  must 
be  taught  there.  What,  then,  may  we  hope  to  accomplish  by 
its  study? 

Part  II. 

The  aims  and  purposes  which  may  be  realized  in  the  study 
of  mathematics  are  determined  by  its  values  to  the  one  who 
studies  it.  These  possible  values  are  almost  without  number. 
We  shall  consider  some  of  the  more  important. 


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1.  Mathematics  Teaches  Logical  Thinking. 

It  is  the  most  effective  means  for  teaching  logical  thinking 
aside  from  the  actual  study  of  logic;  in  this  it  is  unique.  Every¬ 
one  needs  this  training  and  no  other  high  school  study  can  give 
it  so  well.  John  S.  Mill  (“System  of  Logic,”  Bk.  3,  chap.  24, 
sec.  9)  says,  “The  value  of  mathematical  instruction  as  a  prep¬ 
aration  for  those  more  difficult  investigations  consists  in  the 
applicability  not  of  its  doctrines  but  of  its  methods.  Mathe¬ 
matics  will  ever  remain  the  past  perfect  type  of  the  deductive 
method  in  general.”  Benjamin  Pierce  ( American  Journal 
of  Mathematics ,  vol.  4,  p.  97)  says,  “Mathematics  is  the  science 
which  draws  necessary  conclusions.”  The  type  of  reasoning 
most  emphasized  in  mathematics  is  the  deductive.  This  sort 
of  reasoning  is  used  in  other  subjects  and  is  applicable  to  all 
sorts  of  situations  in  life.  Deduction  is  the  process  of  arriving 
at  a  logical  inference  based  on  accepted  premises.  Psycholo¬ 
gists  claim  that  all  thinking  is  problem  solving.  The  process 
as  outlined  by  Professor  Thorndike,  of  Teachers  College,  is  as 
follows : 

1.  A  clear  statement  of  the  goal  aimed  at. 

2.  The  selection  of  enough  and  representative  individual 
facts. 

3.  Their  arrangement  in  such  a  way  as  to  make  the  general 
idea  or  judgment  to  which  they  lead  obvious. 

4.  The  verification  of  the  conclusion  by  an  appeal  to  known 
facts. 

5.  Its  reinforcement  and  clarification  by  exercises  in  applying 
it  to  new  individual  facts. 

It  is  evident  that  the  study  of  mathematics  gives  training  in 
this  process.  The  committee  on  Secondary  Mathematics, 
appointed  by  the  New  England  Association  of  Teachers  of 
Mathematics,  says,  “Whatever  one’s  occupation,  it  will  be  funda¬ 
mentally  important  to  have  acquired  in  youth  the  habit  of  exact, 
orderly  and  logical  thinking,  which,  if  the  experience  of  many 
centuries  of  teaching  can  be  trusted,  is  best  acquired  by  most 
high  school  students  in  mathematics  well  taught.”  Mathe¬ 
matics  is,  then,  of  first  rank  among  high  school  studies  in  teach¬ 
ing  pupils  how  to  think.  Its  fundamental  concepts  are  few  in 
number,  very  simple,  and  lie  close  to  the  experience  of  the 
pupil.  It  should  therefore  be  an  easy  study  for  any  normal 
mind.  John  Locke  (“Conduct  of  the  Understanding”)  says, 
“Would  you  have  a  man  reason  well,  you  must  use  him  to  it 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


119 


betimes.  Exercise  his  mind  in  observing  the  connection  between 
ideas  and  following  them  in  train.  Nothing  does  this  better 
than  mathematics,  which,  therefore,  I  think  should  be  taught 
to  all  who  have  the  time  and  opportunity,  not  so  much  to  make 
them  mathematicians,  as  to  make  them  reasonable  creatures. 
...  In  all  sorts  of  reasoning,  every  simple  argument  should 
be  managed  as  a  mathematical  demonstration.” 

2.  Mathematics  Creates  Self-Confidence. 

This  is  not  the  confidence  that  makes  “fools  step  in  where 
angels  fear  to  tread”;  but  a  confidence  based  on  that  self-knowl¬ 
edge,  and  on  that  self-command  which  give  poise  and  power. 
The  gaining  of  this  confidence  is  a  necessary  part  of  the  early 
education  of  every  individual,  and  it  can  be  accomplished 
better  by  mathematics  than  by  any  other  study.  Other  sub¬ 
jects  depend  on  authority,  and  there  is  usually  a  difference 
between  authorities  and  a  conflict  of  opinions.  The  student 
is  frequently  bewildered.  He  wonders  which  authority  to  ac¬ 
cept,  and  he  doubts  his  ability  to  think  out  a  conclusion  of  his 
own.  He  finds  mathematics  different.  In  mathematics  there 
is  no  such  thing  as  an  outside  authority:  given  certain  premises, 
there  can  be  no  doubt  about  the  results  of  his  reasoning.  Results 
are  either  right  or  wrong  and  they  can  be  checked:  algebra  by 
arithmetical  calculation,  geometry  by  actual  measurement. 
The  reasoning  of  a  student  who  lacks  ability  to  study  mathe¬ 
matics,  or  who  lacks  training  in  it,  is  likely  to  be  of  doubtful 
value — he  is  likely  to  be  uncertain  of  it  himself. 

If  mathematics  has  been  properly  taught  and  thoroughly 
mastered,  one  will  not  place  too  much  dependence  on  the  opin¬ 
ions  of  others.  In  this  way  its  study  will  make  for  a  better 
citizenship.  The  mastery  of  mathematics  must  insure  the  pres¬ 
ence  in  a  man  of  those  qualities,  at  least  some  measure  of  them, 
which  make  leaders.  The  success  of  a  democratic  govern¬ 
ment  depends  on  the  power  of  its  people  to  think  intelligently 
and  to  act  wisely.  Sir  James  Bryce  says,  “It  is  by  the  best 
minds  that  nations  win  and  retain  leadership.  No  pains  can  be 
too  great  that  are  spent  on  developing  such  minds  to  the  finest 
point  of  efficiency.”  It  .’is  equally  true  that  no  pains  are  too 
great  that  will  raise  the  intelligence  of  the  mass  of  the  people 
to  greater  efficiency.  It  is  economy  to  use  mathematics  as  a 
means  to  these  ends.  The  life  of  Lincoln  illustrates  this.^  In 
the  “Life  of  Lincoln,”  Nicolay  and  Hay,  vol.  1,  p.  229,  we  read: 
“It  was  at  this  time  that  he  gave  notable  proof  of  his  unusual 


120 


SCHOOL  SCIENCE  AND  MATHEMATICS 


powers  of  mental  discipline.  His  wider  knowledge  of  men  and 
things,  acquired  by  contact  with  the  great  world,  had  shown 
him  a  certain  lack  in  himself  of  the  power  of  close  and  sustained 
reasoning.  To  remedy  this  defect,  he  applied  himself,  after  his 
return  from  Congress,  to  such  works  upon  logic  and  mathematics 
as  he  fancied  would  be  serviceable.  Devoting  himself  with 
dogged  energy  to  the  task  in  hand,  he  soon  learned  by  heart 
six  books  of  the  propositions  of  Euclid,  and  he  retained  through¬ 
out  life  an  intimate  knowledge  of  the  principles  they  contain.” 
This  is  not  an  isolated  case;  others  give  testimony  to  the  value 
of  the  study  of  mathematics  in  giving  power  to  direct  men  on 
important  issues. 

3.  Mathematics  Cultivates  the  Power  of  Concentration. 

The  success  of  a  student  depends  on  his  early  gaining  of  the 
power  to  concentrate  his  mind  on  a  given  problem.  To  work 
with  the  highest  efficiency  the  mind  must  be  wholly  absorbed 
with  the  task  in  hand,  the  nerves  tense,  and  the  body  in  an 
attitude  that  suggests  attention  and  alertness.  The  easy- 
chair  loafing  method  will  not  lead  to  a  trained  and  developed 
mind.  The  pupil  whose  mind  goes  “wool-gathering”  in  the 
midst  of  important  work,  or  who  is  easily  distracted  by  his 
surroundings,  is  a  failure  as  a  student.  This  power  to  study 
is  developed  by  mathematics  as  by  few  other  studies.  While 
apparent  progress  might  be  made  in  some  studies  without  it, 
concentration  is  vital  in  this  study.  The  New  England  Report 
says,  “The  real  development  of  mankind  lies  in  the  growth 
of  voluntary  attention,  which  is  not  passively  attracted,  but 
turns  actively  to  that  which  is  important,  significant,  and 
valuable  in  itself.  No  one  is  born  with  such  power.  It  has 
to  be  trained  and  educated.  This  great  function  of  education 
is  too  often  neglected.”  We  are  inclined  to  forget  that  real 
progress  is  directly  proportional  to  the  conscious  effort  of  the 
individual.  Prof.  W.  C.  Bagley,  of  Teachers  College,  says, 
“Bricks  cannot  be  made  without  straw,  nor  can  mental  growth 
be  achieved  without  individual  effort  and  individual  sacrifice.” 
Work  is  not  play  and  the  child  must  distinguish  between  the 
two.  Something  must  be  done  besides  amusing  our  pupils  in 
class.  There  is  something  more  important  than  catering  to  the 
child’s  wishes.  Sometimes  he  may  have  duties  to  perform  that 
he  will  not  like.  He  needs  the  training  that  such  effort  will 
give,  since  he  must  learn  to  adjust  himself  to  environment, 
and  to  the  comfort  and  wishes  of  others  as  well  as  of  himself. 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


121 


Spencer  defined  the  educated  being  as  one  who  did  what  he 
ought,  when  he  ought,  whether  he  wanted  to  or  not.  The 
pupil  must  learn  that  the  world  does  not  revolve  about  himself; 
and  we  as  teachers  should  recognize  the  fact  that  a  pupil  does 
not  find  himself  educated  without  knowing  how  it  happened. 
Mathematics  has  the  power  to  develop  concentration;  and  it 
is  our  duty  to  furnish  adequate  motive  for  the  effort  that  will 
accomplish  this  result.  Professor  R.  E.  Moritz  (University  of 
Washington)  says,  “That  mathematics  is  the  most  efficient 
agency  for  acquiring  the  power  of  quick  attention  and  prolonged 
concentration  of  mind  has  never  been  seriously  questioned  by 
competent  critics.”  Prof.  Gonzales  Lodge,  of  Teachers  College 
(Address  before  the  faculty  of  Teachers  College,  February  8, 
1917)  says,  “It  is  apparently  becoming  more  and  more  a  cardinal 
doctrine  of  the  shallow  thinkers  of  the  present  day  and  genera¬ 
tion  that  the  mind  of  man  cannot  be  trained;  that  the  only 
thing  that  can  be  trained  is  the  hand.  ...  It  seems  to  me 
axiomatic  that  mental  training  to  be  valuable  must  involve 
effort.  ...  I  mean  that  kind  of  effort,  necessary  in  the  devel¬ 
opment  of  the  attitude  of  mind  that  faces  a  problem  squarely, 
which  goes  to  work  at  it  in  detail,  which  analyzes  it  with  care  and 
exactness;  which  expresses  the  results  of  this  analysis  with  the 
same  care  and  exactness.  The  mind  that  can  do  that  is  cer¬ 
tainly  a  trained  mind,  and  the  benefits  of  such  training  are 
available  in  every  walk  of  life.” 

4.  Mathematics  Demands  Originality  in  Its  Study. 

It  is  probable  that  much  of  our  school  work  stifles  initiative 
and  originality,  if  it  does  not  kill  these  outright,  in  attempting 
to  fashion  large  class  groups  according  to  one  mould.  This 
deplorable  condition  is  aggravated  by  the  excessive  dependence 
on  the  text  and  the  memory  work  so  common  in  all  studies. 
If  the  pupil  can  make  a  show  of  learning  by  repeating  other 
people’s  ideas  he  is  passed  on  and  the  teacher  is  counted  a  suc¬ 
cess.  Mathematics,  properly  taught,  and  especially  in  its 
application  to  well  chosen  practical  problems,  requires  inde¬ 
pendent  thought  and  judgment.  The  pupil  is  able  to  realize 
all  the  joy  of  individual  discovery  and  achievement,  which  is  in 
itself  a  sufficient  motive  for  the  study  of  any  subject.  His  new 
consciousness  of  independence  and  of  power  is  an  inspiration 
and  a  delight  to  a  reasonably  keen-minded  student  and  is  fre¬ 
quently  a  stimulus  to  a  slower  mind.  Mathematics  is  usually 
the  child’s  first  introduction  to  the  possibility  of  independent 


122 


SCHOOL  SCIENCi 1  AND  MATHEMATICS 


thinking;  and  at  a  time  when  the  independence  of  maturity- 
should  be  developing,  it  is  especially  suited  to  his  needs.  How¬ 
ever,  it  sometimes  happens  that  a  child’s  ability  to  remember 
and  to  repeat  glibly  is  mistaken  for  ability  to  think.  Conse¬ 
quently  a  1  ‘bright”  pupil  fails  in  mathematics  and  the  subject 
or  the  teacher  is  condemned  by  disappointed  parents.  To  say 
that  the  study  of  mathematics  does  not  develop  originality, 
or  to  teach  it  as  a  thing  to  be  learned  by  rote  and  to  be  applied 
by  machine  methods,  is  to  debase  it  to  the  rank  of  a  mere  tool 
and  create  the  possibility  that  a  show  of  learning  may  be  made 
by  a  mere  exercise  of  the  memory.  There  can  be  little  gain 
for  the  pupil  in  either  case.  J.  J.  Sylvester  (“Mathematical 
Papers”)  says,  “As  the  prerogative  of  natural  science  is  to 
cultivate  a  taste  for  observation,  so  that  of  mathematics  is, 
almost  from  the  starting  point,  to  stimulate  the  faculty  of  inven¬ 
tion.” 

Again,  some  students  may  possess  unusual  abilities  in  mathe¬ 
matics  or  in  related  fields.  A  lack  of  a  knowledge  of  algebra 
and  geometry  in  early  years  may  mean  that  these  will  remain 
forever  buried  and  that  the  race  may  be  deprived  of  an  important 
possible  contribution  to  its  progress.  Even  unusual  difficulties 
with  these  subjects  is  not  always  a  sufficient  reason  for  the 
abandonment  of  them.  Dr.  Smith  ( Mathematics  Teacher , 
March,  1913)  speaks  of  some  of  the  world’s  greatest  mathe¬ 
maticians  as  being  unpromising  in  early  years.  Florian  Cajori 
(“A  History  of  Mathematics,”  p.  201)  says  of  Newton,  “At 
first  he  seems  to  have  been  very  inattentive  to  his  studies  and 
of  very  low  rank  in  school.”  The  study  of  mathematics  will, 
then,  draw  out  the  individual  powers  of  the  pupil  and  enable 
him  to  “find”  himself.  It  furnishes  the  best  measuring  rod 
for  a  pupil’s  abilities  and  needs. 

5.  Mathematics  Trains  in  the  Precise  Use  of  English. 

Teachers  of  algebra  and  geometry  know  that  students  have 
the  greatest  difficulties  with  translation  problems  in  algebra 
and  with  originals  in  geometry.  Most  of  the  trouble  is  due  to 
inability  to  read  intelligently.  Reading,  of  course,  means  getting 
the  thought.  Much  of  the  pupil’s  reading  up  to  this  point  has 
been  so  simple  as  to  require  little  effort  to  get  the  meaning,  or 
it  may  have  been  the  mere  repetition  of  words.  The  pupil 
must  be  taught  to  dig  below  the  surface,  to  properly  balance 
statements  and  to  get  their  real  meaning.  The  pupil  must 
also  learn  to  express  himself  in  a  concise  and  forceful  manner. 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


123 


Definitions  must  be  clearly  and  accurately  stated,  not  a  word 
too  many,  not  a  word  lacking,  just  the  right  word  for  each 
idea.  Hypotheses  must  be  stated  exactly  and  kept  distinct 
from  conclusions  to  be  reached*  All  of  this  is  hard  work  and 
requires  much  patience,  but  it  is  a  worthy  effort  for  both  teacher 
and  student.  English  teachers  who  have  had  the  opportunity 
to  observe  are  emphatic  in  their  approval  of  mathematics  as 
an  aid  in  the  mastery  of  English,  correcting  slovenliness  and 
inaccuracy. 

Much  useless  discussion  and  controversy  would  be  avoided 
if  the  precise  use  of  English  were  made  a  more  direct  aim  in 
the  study  of  mathematics.  Professor  Moritz  says,  “In  mathe¬ 
matics,  therefore,  the  student  can  be  brought  to  recognize 
the  absolute  necessity  of  mastering  the  meaning  of  words  pre¬ 
liminary  to  their  use  as  a  vehicle  of  thought.  Half  of  the  mis¬ 
understandings  and  futile  controversies  in  active  fife  arise  from 
ambiguity  in  the  use  of  words.  .  .  .  Mathematics  has  come  to 
be  accepted  as  the  synonym  for  exactness,  clearness,  certainty.” 
Perhaps  in  no  other  field  is  this  training  more  essential  than  in 
the  practice  of  law.  Demonstrating  before  a  class  and  answer¬ 
ing  questions  from  teacher  and  fellow  pupils  gives  self-possession, 
ability  to  think  on  one’s  feet,  and  the  power  to  adjust  in  proper 
order  the  important  parts  of  any  problem,  giving  to  each  its 
proper  weight,  so  that  desired  ends  may  be  attained.  The 
pupil  thus  acquires  ability  to  discuss  a  problem  intelligently 
before  a  critical  audience.  Thomas  Jefferson  was  of  the  opinion 
that,  “mathematical  reasoning  and  deductions  are  a  fine  prepara¬ 
tion  for  investigating  the  abstruse  speculations  of  the  law.” 

6.  Mathematics  Trains  in  Accuracy. 

Probably  no  other  subject  demands  this  quality  to  so  great 
a  degree.  There  is  no  opportunity  to  cloak  errors  with  results 
“nearly”  right.  If  approximate  results  are  sought  the  limits 
for  the  errors  are  known.  Pupils  can  judge  for  themselves  and 
make  their  own  corrections.  Again,  carelessness  in  securing 
data  or  in  the  drawing  of  a  figure  may  vitiate  the  whole  prob¬ 
lem,  no  matter  how  perfect  the  reasoning.  Furthermore,  neat¬ 
ness  is  an  essential  part  of  accuracy  and  both  should  be  conscious 
aims  of  both  teacher  and  pupil.  They  should  be  emphasized 
more  than  at  present  in  our  teaching  of  mathematics. 

(To  be  continued) 


208 


SCHOOL  SCIENCE  AND  MATHEMATICS 


VALID  AIMS  AND  PURPOSES  FOR  THE  STUDY  OF 
MATHEMATICS  IN  SECONDARY  SCHOOLS. 

By  Alfred  Davis, 

Francis  W.  Parker  School, 

330  Webster  Ave.,  Chicago. 

Chairman  of  a  Committee  of  the  Mathematics  Club  of  Chicago 
Appointed  to  Investigate  This  Topic. 

(Continued  from  the  February  number.) 

7.  Mathematics  Gives  Ability  to  Handle  a  Tool,  Essential  in 
Much  of  Life’s  Work. 

Failure  to  recognize  this  often  makes  trouble  for  the  student 
later.  Prof.  A.  It.  Crathorne,  of  the  University  of  Illinois,  says, 
“we  have  in  the  University  of  Illinois  graduate  students  in 
agriculture  who  find  themselves  under  the  necessity  of  delaying 
the  work  in  which  they  are  directly  interested  in  order  to  study 
the  freshman  algebra  that  they  find  essential  to  the  study  of 
their  problems”  (School  Science  and  Mathematics,  vol. 
16,  p.  420).  And  further,  “The  utility  of  algebra  as  a  medium 
of  expression  is  on  the  increase  as  surely  as  we  are  gaining  more 
exact  scientific  knowledge”  (Ibid.).  The  list  of  courses  for 
which  mathematics  is  an  essential  prerequisite,  as  given  by  a 
committee  of  this  club  (School  Science  and  Mathematics, 
vol.  16,  pp.  610-611)  is  as  follows:  scientific  agriculture,  engi¬ 
neering,  physics,  chemistry,  art  (drawing,  designing,  architec¬ 
ture,  modeling,  life  and  still  life  drawing,  handicraft),  pharmacy, 
dentistry,  navigation,  astronomy,  naval  and  military  engineer¬ 
ing,  domestic  science,  insurance,  forestry,  commerce  and  ad¬ 
ministration,  railway  administration,  political  economy,  sociology, 
hygiene,  sanitation,  education,  medicine,  and  journalism. 
Law  and  theology  are  included  with  some  reservations.  It 
is  not  difficult  to  see  that  a  knowledge  of  algebra  and  geometry 
is  absolutely  essential  for  some  of  these  courses,  while  in  the 
others  it  has  important  uses,  particularly  the  formula,  the  graph, 
and  the  equation  of  elementary  algebra. 

The  man  of  ordinary  education  needs  a  knowledge  of  mathe¬ 
matics  to  appreciate  everyday  literature  on  many  topics.  Read¬ 
ing  the  many  books  and  magazines  relating  to  the  automobile, 
the  aeroplane,  progress  in  science,  or  the  war,  without  this  as  a 
foundation,  is  like  reading  about  various  places  of  interest 
without  a  knowledge  of  place  geography.  Prof.  S.  G.  Barton, 
of  the  Flower  Observatory,  University  of  Pennsylvania  ( Science , 
vol.  40,  p.  697)  says  that  in  the  Encyclopedia  Brittanica  (11th 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


209 


ed.)  there  are  104  articles  which  use  the  calculus,  of  which 
about  one-fourth  are  pure  mathematics.  Some  of  these  articles 
are:  clock,  heat,  lubrication,  map,  power  transmission,  ship 
building,  sky,  steam  engine,  etc.  There  is  much  greater  need 
for  a  knowledge  of  algebra  and  geometry  in  everyday  reading. 
Prof.  D.  E.  Smith,  of  Columbia,  says,  “Of  the  necessity  for 
knowing  number  relations  there  can  be  no  question,  but  fifty 
years  ago  one  might  well  have  cried  the  slogan  abroad  from  the 
housetops,  ‘Will  anyone  tell  me  why  the  girl  should  study 
algebra?’  Today  a  person  would  sadly  feel  his  ignorance  if  he  or 
she  had  to  look  with  lack-lustre  eyes  upon  a  simple  formula  such 
as  may  be  found  in  Popular  Mechanics,  Motor ,  the  Scientific 
American,  an  everyday  article  on  astronomy,  a  boy’s  manual 
on  the  airplane,  or  any  one  of  hundreds  of  articles  in  our  popular 
encyclopedias.  These  needs  come  not  only  within  the  purview 
of  the  boy;  they  are  even  more  apparent  in  the  case  of  the 
girl,  she  who  is  to  have  the  direction  of  the  education  of  the 
generation  next  to  come  upon  the  stage  of  action.  Each  must 
know  the  shorthand  of  the  formula,  and  the  meaning  of  a  simple 
graph,  of  a  simple  equation,  and  of  a  negative  number,  or  else 
must  feel  the  stigma  of  ignorance  of  the  common  things  that 
the  educated  world  talks  about  and  reads  about.”  T.  C.  Record 
May,  1917.) 

It  is  not  enough  that  we  aim  at  the  application  of  the  results 
which  others  have  worked  out.  We  must  go  deeper  than  that. 
An  intelligent  use  of  mathematics  demands  a  knowledge  of  the 
subject.  It  is  the  testimony  of  a  teacher  in  a  correspondence 
school  that  a  student  can  in  a  few  lessons  learn  the  application 
of  a  formula  to  a  particular  situation;  but  that  when  a  new 
situation  arises  there  is  no  resourcefulness  to  meet  the  new 
need.  The  student  has  not  mastered  the  subject.  His  place 
in  the  industrial  world  must  be  that  of  a  machine.  A.  R.  Forsyth, 
President  of  the  British  Association  for  the  Advancement  of 
Science,  Sec.  A,  says  regarding  the  Perry  movement,  “Some¬ 
thing  has  been  said  about  the  use  of  mathematics  in  physical 
science,  the  mathematics  being  regarded  as  a  weapon  forged  by 
others,  and  the  study  of  the  weapon  being  completely  set  aside. 
I  can  only  say  that  there  is  danger  of  obtaining  untrustworthy 
results  in  physical  science  if  only  the  results  of  mathematics 
are  used:  for  the  person  so  using  the  weapon  can  remain  unac¬ 
quainted  with  the  conditions  under  which  it  can  be  rightly 
applied.  .  .  .  The  results  are  often  correct,  sometimes  incorrect; 


210 


SCHOOL  SCIENCE  AND  MATHEMATICS 


the  consequence  of  the  latter  class  of  cases  is  to  throw  doubt 
upon  all  the  applications  of  such  a  worker  until  a  result  has  been 
otherwise  tested.  Moreover,  such  a  practice  in  the  use  of 
mathematics  leads  a  worker  to  a  mere  repetition  in  the  use  of 
familiar  weapons;  he  is  unable  to  adapt  them  with  any  confidence 
when  some  new  set  of  conditions  arises  with  a  demand  for  a  new 
method:  for  want  of  adequate  instruction  in  the  forging  of  the 
weapon,  he  may  find  himself,  sooner  or  later  in  the  progress 
of  his  subject,  without  any  weapon  worth  having.”  .  .  .  “The 
witness  of  history  shows  that,  in  the  field  of  natural  philosophy, 
mathematics  will  furnish  the  more  effective  assistance  if,  in  its 
systematic  development,  its  courses  can  freely  pass  beyond  the 
ever-shifting  domain  of  use  and  application.”  (Perry’s  “Teach¬ 
ing  of  Mathematics,”  p.  36,  and  Nature ,  vol.  56,  p.  377.) 

The  study  of  mathematics  is  useful  in  giving  set  and  balance 
to  one’s  life.  The  more  widespread  its  study  and  the  resulting 
mathematical  sense  the  less  opportunity  there  will  be  for  dis¬ 
honest  practices  by  the  unscrupulous  man  of  affairs;  a  sort  of 
intuitive  sense  of  the  correctness  of  business  transactions  will 
often  prevent  errors  and  losses;  the  elements  of  chance  and  luck 
will  play  a  less  important  part  in  our  affairs  and  superstition,  the 
stronghold  of  ignorance  blocking  the  way  of  progress,  will  be 
demolished. 

8.  Mathematics  Gives  Training  in  the  Use  of  a  Symbolic  Lan¬ 
guage. 

Much  of  the  world’s  work  is  done  by  the  use  of  symbols.  They 
are  the  tools  for  rapid  thinking  and  writing.  The  progress 
mankind  has  made  would  be  impossible  without  them.  We  are 
convinced  of  this  when  we  think  of  carrying  on  the  simple  opera¬ 
tions  of  multiplication  and  division  without  the  use  of  figures. 
Newton’s  law  of  gravitation:  F  =  GMm/D2,  where  F  is 
the  force  acting  between  two  bodies,  M  and  m  are  the 
masses  of  the  bodies,  D  the  distance  between  their  centers,  and 
G  the  constant  of  gravitation,  would  be  cumbersome  indeed  if  it 
were  necessary  to  say  it  the  long  way  in  order  to  use  it.  By  the 
use  of  the  algebraic  symbols  the  whole  story  is  gotten  at  a 
glance,  and  in  a  form  convenient  for  application  to  other  prob¬ 
lems.  There  is  economy  of  mental  effort  as  well  as  of  time.  The 
importance  of  symbols  is  further  illustrated  in  the  fact  that 
Newton  chose  a  clumsy  system  of  notation  for  the  calculus  while 
Leibnitz  chose  the  present  system,  which  is  much  better.  The 
English  adopted  Newton’s  method  and  so  fell  far  behind  the 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


211 


mathematicians  of  the  continent  in  development  and  applica¬ 
tion  of  the  calculus.  A  reprint  from  the  Engineering  Supple¬ 
ment  of  the  London  Times ,.  June  19,  1910,  says,  “The  extent  to 
which  mathematics  is  capable  of  exact  prediction  depends  on 
expressing  the  problem  in  mathematical  language.  The  greater 
ability  of  engineers  of  today  to  translate  problems  into  this 
language  has  led  to  an  increasing  number  of  successful  inven¬ 
tions. The  interpretation  and  application  of  formulae  is  nec¬ 
essary  in  the  reading  of  all  sorts  of  current  literature,  and  in 
the  application  of  general  principles  to  all  sorts  of  human  effort. 
Algebra  and  geometry  offer  unequaled  opportunity  for  the 
mastery  and  use  of  symbols,  since  the  solution  of  problems 
requires  constant  translation  back  and  forth.  Dr.  Smith  ( T .  C. 
Record ,  May,  1917),  “One  merit  of  mathematics  no  one  can 
deny — it  says  more  in  fewer  words  than  any  other  science  in 
the  world.”  {The  Nation,  vol.  33,  p.  426),  “The  human  mind 
has  never  invented  a  labor-saving  machine  equal  to  algebra.” 
Lack  of  training  in  the  use  of  symbols  fixes  very  narrow  limita¬ 
tions  to  one’s  life. 

The  graph  has  become  one  of  the  most  important  symbols 
in  modern  life.  The  meaning  and  use  of  the  graph  cannot  be 
properly  taught  apart  from  algebra.  We  see  at  a  glance  the  re¬ 
lationship  of  two  interdependent  variables  at  any  stage.  Prof. 
Crathorne  (School  Science  and  Mathematics,  vol.  16,  pp. 
423-4)  says,  “The  world  is  full  of  variables  which  depend  on 
other  variables,  presenting  to  us  the  problem  of  finding  out 
and  exhibiting  the  manner  of  dependence.  The  office  of  the 
graphical  methods  of  algebra  is  to  exhibit  this  dependence  to 
the  eye,  and  not,  as  many  textbooks  would  imply,  merely  to  aid 
in  the  solution  of  equations.  ...  In  a  recent  book  for  work¬ 
ingmen  in  shop  mechanics,  a  full  page  is  devoted  to  an  explana¬ 
tion  of  the  stretch  of  copper  wire  for  different  loads.  The  author, 
no  doubt  realizing  the  vagueness  of  his  explanation,  then  clear¬ 
ly  sums  up  by  a  graph  with  three  lines  of  English  under  it.” 
The  graph  has  become  well-nigh  indispensable  in  presenting 
statistics.  Much  labor  is  necessary  to  get  facts  from  numerical 
data,  while  the  graph  gives  the  same  information  almost  without 
effort  and  presents  a  picture  to  the  mind  that  is  easy  to  recall. 
The  graph  makes  the  function  concept  clear  to  the  pupil.  He 
is  able  to  understand  the  meaning  and  something  of  the  impor¬ 
tance  of  “x  is  a  function  of  y.”  When  the  graph  is  subject  to  a 
known  law,  as  represented  by  an  equation,  it  is  a  necessary  step 
to  the  study  of  advanced  mathematics. 


212 


SCHOOL  SCIENCE  AND  MATHEMATICS 


Regarding  the  saving  of  mental  energy  by  a  knowledge  of 
mathematics,  and  by  a  mastery  of  the  use  of  its  symbols,  Prof. 
Chas.  H.  Judd,  of  the  University  of  Chicago  (“The  Psychology 
of  High  School  Subjects,”  p.  131)  says,  “No  student  will  know 
what  mathematics  is  until  he  realizes  the  great  economy  of 
mental  energy  which  this  form  of  experience  makes  possible.” 

9.  Mathematics  Develops  the  Imagination. 

G.  St.  L.  Carson,  in  “Mathematical  Education,”  p.  41,  says, 
“The  operations  and  processes  of  mathematics  are  in  practice 
concerned  at  least  as  much  with  creations  of  the  imagination 
as  with  the  evidence  of  the  senses.”  F.  J.  Herbart  says,  “The 
great  science  (mathematics)  occupies  itself  at  least  just  as  much 
with  the  power  of  imagination  as  with  the  power  of  logical  con¬ 
clusion.”  Prof.  H.  H.  Horne,  of  Dartmouth  College,  says, 
“Apart  from  its  manifold  applications,  mathematics  is  the 
inevitable  disciplinary  element  in  the  curriculum.  It  trains  in 
the  habit  of  logical  and  symbolic  thinking,  of  precision  and  con¬ 
centration,  and  it  develops  the  imagination.”  The  concepts 
of  space  and  number  relationships  are  fundamental  in  educa¬ 
tion.  Col.  F.  W.  Parker  (“Talks  on  Pedagogics,”  p.  50,  etc.) 
says,  “I  think  we  can  truthfully  say  that  form  is  the  supreme 
manifestation  of  energy,  and  without  a  knowledge  of  form  and 
without  the  power  to  judge  form  with  some  degree  of  accuracy, 
there  can  be  no  such  thing  as  educative  knowledge.  .  .  .  Form 
and  number  are  modes  of  judging  and  are  necessary  to  a  knowl¬ 
edge  of  the  external  world.  .  .  .  The  study  of  form  and  geometry 
are  of  fundamental,  intrinsic  importance  in  education.”  In  the 
study  of  mathematics,  images  of  one,  two,  and  three  d  mensions 
are  constantly  before  the  mind.  At  first,  objects  and  drawings 
are  used  to  give  clear  pictures,  but  the  student  soon  learns  to 
depend  on  the  imagination  to  reproduce  the  images  and  to 
frame  new  relationships.  This  is  especially  true  in  the  study  of 
geometry.  In  the  discussion  of  a  problem,  all  the  possibilities 
of  a  given  case  must  pass  in  order  before  the  mind.  Surely  the 
imagination  is  used  and  cultivated  in  the  study  of  mathematics. 

A  few  well-selected  problems  in  physics  and  astronomy, 
taught  with  appreciation  by  the  teacher,  will  aid  in  the  cultiva¬ 
tion  of  the  imagination  and  broaden  the  life  of  the  pupil.  These 
may  relate  to  the  velocity  of  light  and  the  length  of  its  wave; 
the  relative  sizes  of  the  planets  and  stars  and  how  these  are 
measured;  the  distance  of  the  planets  and  stars  and  the  meaning 
of  “light  years”;  etc.  Experience  shows  that  problems  relating 


MATHEMATICS  IN  SECONDARY  SCHOOLS  213 

to  such  topics  are  fascinating  to  the  student.  Carson  (“Math. 
Ed.,”  p.  10),  says,  “One  of  the  few  really  certain  facts  about  the  / 
juvenile  mind  is  that  it  revels  in  the  exploration  of  the  un¬ 
known.”  Astronomy,  in  particular,  leads  the  mind  to  the  thresh¬ 
old  of  the  unknown,  and  exposes  it  to  the  Infinite.  Who  can 
be  little,  or  narrow,  or  prejudiced,  if  his  imagination  has  been 
inspired  by  mathesis ! 

10.  Mathematics  Leads  to  a  Knowledge  and  an  Appreciation  of 
the  Foundations  of  Science. 

All  science  is  ultimately  mathematical  in  its  methods;  the 
more  completely  it  is  developed  the  more  mathematical  a 
science  becomes.  Mathematics  enables  us  to  apply  accepted 
laws  to  Nature’s  problems.  In  this  way  Newton’s  and  Kepler’s 
laws  have  been  established  and  have  been  made  to  extend  to 
almost  infinite ‘reaches  into  space  and  to  unfold  the  mysteries 
of  a  universe  of  which  we  are  an  infinitesimal  part.  Astronomy 
was  astrology  until  mathematics  released  it  and  it  became  a 
science.  But  even  astrology  depended  somewhat  on  mathe¬ 
matics.  Sir  John  Herschel  (“Outlines  of  Astronomy,”  Intro¬ 
duction,  Sec.  7)  says,  “Admission  to  its  sanctuary  (astronomy) 
and  to  the  privileges  of  a  votary  is  only  to  be  gained  by  one 
means,  sound  and  sufficient  knowledge  of  mathematics,  the  great 
instrument  of  all  exact  inquiry ,  without  which  no  man  can  ever 
make  such  advances  in  this  or  any  other  of  the  higher  departments 
of  science  as  can  entitle  him  to  form  an  independent  opinion  on 
any  subject  of  discussion  within  their  range.”  It  is  through 
mathematics  that  we  are  gaining  knowledge  of  molecules, 
atoms,  electrons,  ions;  of  the  wonderful  changes  that  are  occur¬ 
ring  in  these  and  of  the  laws  that  govern  them.  Mathematics 
has  well-nigh  unlocked  the  secret  of  matter  itself.  Indeed, 
we  could  know  little  of  chemistry,  physics,  or  of  any  other 
science  were  it  not  for  the  aid  of  mathematics;  witness  the  fol¬ 
lowing  testimony:  Roger  Bacon  (“Opus  Majus”)  “Mathematics 
is  the  gate  and  the  key  of  the  sciences.  .  .  .  Neglect  of  mathe¬ 
matics  works  injury  to  all  knowledge,  since  he  who  is  ignorant 
of  it  cannot  know  the  other  sciences  or  the  things  of  this  world. 
And  what  is  worse,  men  who  are  thus  ignorant  are  unable  to 
perceive  their  own  ignorance  and  so  do  not  seek  a  remedy.”  A 
Kant,  “A  natural  science  is  a  science  only  in  so  far  as  it  is  mathe¬ 
matical.”  Laplace,  “All  the  effects  of  nature  are  only  mathe¬ 
matical  results  of  a  small  number  of  immutable  laws.”  W. 
W.  R.  Ball  (“History  of  Mathematics,”  p.  503)  “The  advance 


214 


SCHOOL  SCIENCE  AND  MATHEMATICS 


in  our  knowledge  of  physics  is  largely  due  to  the  application  to 
it  of  mathematics,  and  every  year  it  becomes  more  difficult 
for  an  experimenter  to  make  any  mark  in  the  subject  unless  he 
is  also  a  mathematician.”  Comte,  “All  scientific  education 
that  does  not  begin  with  mathematics  is  defective  at  its  founda¬ 
tion.  ...  In  mathematics  we  find  the  primitive  source  of 
rationality;  and  to  mathematics  must  biologists  resort  for  means 
to  carry  on  their  researches.”  J.  F.  Herbart,  “It  is  not  only 
possible  but  necessary  that  mathematics  be  applied  to  psychol¬ 
ogy;  the  reason  for  this  necessity  lies  briefly  in  this:  that  by 
no  other  means  can  be  reached  that  which  is  the  ultimate  aim 
of  all  speculation,  namely  conviction .”  Novalis,  “All  historic 
science  tends  to  become  mathematical.  Mathematical  power 
is  classifying  power.”  Prof.  A.  Voss,  of  the  University  of  Munich, 
in  a  lecture  in  1903  (quoted  by  T.  E.  Mason  of  Purdue  in  Science , 
December  15,  1916)  said,  “Our  entire  present  civilization,  as 
far  as  it  depends  upon  the  intellectual  penetration  of  nature, 
has  its  real  foundation  in  the  mathematical  sciences.”  Prof. 
Thos.  E.  Mason  {Science,  December  15,  1916),  “Can  you  realize 
what  would  happen,  just  what  stage  of  civilization  we  should 
be  in,  if  all  that  is  developed  by  the  use  of  mathematics  could 
be  removed  from  the  world  by  some  magic  gesture?  Every 
branch  of  physics  makes  use  of  mathematics;  chemistry  is  not 
free  from  it;  engineering  is  based  on  its  development;  sociology, 
economics,  and  variation  in  biology  make  use  of  statistics  and 
probability.  Our  skyscrapers  must  disappear;  our  great  bridges 
and  tunnels  must  be  removed;  our  transportation  systems,  our 
banking  systems,  our  whole  civilization,  indeed,  must  step 
back  many  centuries.”  The  student  who  leaves  high  school 
without  a  knowledge  of  the  importance  of  mathematics  in  science 
has  a  serious  lack.  Problems  in  algebra  and  geometry  should, 
when  convenient,  relate  to  the  various  sciences.  If  the  text 
does  not  furnish  such  problems  the  teacher  should  provide 
them. 

11.  Mathematics  Should  Be  Appreciated  as  One  of  the  Great¬ 
est  Achievements  of  the  Human  Intellect. 

In  this  respect  there  is  the  same  reason  for  studying  mathe¬ 
matics  as  for  studying  literature,  language,  art,  or  history,  for 
it  is  only  as  we  learn  to  appreciate  the  greatest  in  man’s  efforts, 
achievements,  and  aims  that  we  can  have  the  proper  ideals  and 
purposes  in  our  lives.  It  is  only  through  this  knowledge  and 
appreciation  that  one  is  able  to  take  an  intelligent  part  in  the 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


215 


world  and  in  the  age  in  which  one  lives;  that  one  can  be  a  live 
force  and  not  an  encumbrance  in  the  world.  The  path  by  which 
men  have  traveled  is  shown  by  history;  literature  and  art  beau¬ 
tify  its  borders;  language  furnishes  the  bond  of  unity;  but 
science,  including  mathematics,  is  the  pavement  and  even 
provides  the  light  by  which  they  walk.  Indeed,  mathematics 
is  the  greatest  of  the  sciences.  Hermann  Hankel  says,  “In 
most  sciences  one  generation  tears  down  what  another  has 
built  and  what  one  has  established  another  undoes.  In  mathe¬ 
matics  alone  each  generation  builds  a  new  story  to  the  old 
structure.”  The  earth  as  the  center  of  the  universe,  the  cor¬ 
puscular  theory  of  light,  the  indestructibility  of  the  atom, 
these  and  many  other  theories,  at  one  time  considered  funda¬ 
mental,  are  now  fit  only  for  the  intellectual  museum;  but  the 
contributions  to  mathematics  endure.  Dr.  Smith  (T.  C.  Record , 
May,  1917)  says,  in  speaking  of  the  theorem  of  Pythagoras, 
“Before  Mars  was,  or  the  earth,  or  the  sun,  and  long  after  each 
has  ceased  to  exist,  there  and  here  and  in  the  most  remote 
regions  of  stellar  space,  the  square  on  the  hypotenuse  was 
and  is  and  ever  shall  be  equivalent  to  the  sum  of  the  squares  on 
the  sides.  All  our  little  theories  of  life,  all  our  childish  specula¬ 
tions  as  to  death,  all  our  trivial  bickerings  of  the  schools — all 
these  are  but  vanishing  motes  in  the  sunbeam  compared  with 
the  double  eternity,  past  and  future,  of  such  a  truth  as  this.” 
Surely  we  can  appreciate  Laisant  when  he  says,  “Mathematics  is 
the  most  marvelous  instrument  created  by  the  human  mind 
for  the  discovery  of  truth”;  and  Leibnitz,  who  says,  “Mathe¬ 
matics  is  the  glory  of  the  human  mind.” 

Do  we  aim,  as  teachers,  to  give  this  appreciation  of  mathe¬ 
matics?  It  can  be  accomplished  only  by  the  study  of  mathe¬ 
matics,  and  not  by  a  course  about  mathematics,  as  some  have 
suggested.  As  well  expect  nourishment  from  talking  about 
food  as  to  expect  knowledge  and  appreciation  of  mathematics 
from  talking  about  it.  More  importance  ought  to  be  given  the 
history  of  mathematics  in  our  teaching.  Much  inspiration 
and  enthusiasm  can  be  gained  from  the  lives  of  great  mathe¬ 
maticians.  And  at  appropriate  times  there  should  be  given  the 
student  a  prospective  view  of  the  richness  and  beauties  of  the 
subject  to  be  realized  by  advanced  study. 

12.  Mathematics  May  Make  An  Important  Contribution  to 
the  Aesthetic ,  Moral ,  and  Religious  Life  of  the  Individual. 

Henri  Poincar6  ( Annual  Report ,  Smithsonian  Institution, 


216 


SCHOOL  SCIENCE  AND  MATHEMATICS 


1909)  says  that  mathematics  has  aesthetic  value  in  the  feeling 
of  elegance  in  a  solution  or  demonstration;  in  the  harmony 
among  parts,  their  happy  balancing,  and  their  symmetry. 
The  feeling  of  elegance  may  come  from  unexpected  associations 
and  kinships  among  things.  The  sense  of  beauty  is  bound  up 
with  the  economy  of  thought.  We  have  seen  high  school  pupils 
fascinated  by  the  application  of  the  binomial  theorem;  by  the 
power  and  elegance  of  an  algebraic  solution;  or  by  the  Golden 
Section  and  other  geometrical  constructions.  They  are  de¬ 
lighted  with  beauties  in  nature  and  art  that  are  revealed  for 
the  first  time  through  the  study  of  algebra  and  geometry. 
Prof.  S.  G.  Bartoij  ( Science ,  vol.  40)  says,  “Beauty  is  con¬ 
cealed  by  ignorance  of  the  mathematics  necessary  for  its  inter¬ 
pretation.  The  student  of  mathematics  will  see  that  of  which 
the  untutored  mind  has  no  conception,  because  lying  beyond 
its  comprehension.  .  .  .  One  of  nature’s  demands  in  which 
she  is  inexorable  is  a  study  of  higher — the  highest — mathematics. 
The  interpretation  of  her  laws  requires  it. 

“The  massive  bridge  once  wonderful  because  of  its  enormous 
size,  when  its  principles  of  construction  are  understood,  be¬ 
comes  a  thing  of  beauty,  a  wonderful  monument  to  the  intel¬ 
lects  of  the  designer  and  the  constructor.  The  great  tunnels, 
turbines,  subways,  are  changed  to  objects  of  wonder,  to  those 
capable  of  understanding  the  difficulties  overcome  in  their 
construction.  The  stars  in  the  universe  above,  which  nightly 
dissipate  some  of  their  light  upon  the  earth,  bespeak  their 
Creator’s  glory  in  voices  but  faintly  heard  by  those  whose 
training  does  not  enable  them  to  comprehend  the  reign  of  law 
there  prevailing.  To  such  an  one  The  heavens  declare  the 
glory  of  God’  in  a  more  real  and  exalted  sense.”  Thus  mathe¬ 
matics  literally  opens  a  new  earth  and  a  new  heaven  to  us. 
It  unlocks  for  us  the  “music  of  the  spheres”;  it  reveals  the 
thoughts  of  the  Eternal. 

The  study  of  mathematics  leads  to  clear  thinking;  to  honest 
and  patient  effort;  to  reverence  for  truth;  and  must,  therefore, 
have  a  large  place  in  the  building  of  character.  B.  F.  Finkel 
says,  “Mathematics  is  the  very  embodiment  of  truth.  No 
true  devotee  of  mathematics  can  be  dishonest,  untruthful, 
unjust.  Because,  working  with  that  which  is  true,  how  can 
one  develop  in  himself  that  which  is  exactly  opposite?  Mathe¬ 
matics,  therefore,  has  ethical  as  well  as  educational  value.” 
Prof.  H.  E.  Hawkes,  of  Columbia  ( Mathematics  Teacher,  March, 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


217 


1913)  says,  “What  is  simplest  and  most  beautiful  in  the  domain 
of  pure  mathematics  too  often  corresponds  to  the  facts  of  na¬ 
ture  to  be  accidental.  I  contend  that  it  is  our  privilege  to  point 
out,  at  every  possible  turn,  this  coordination  of  number  and 
form,  of  formula  and  physical  law,  of  unity  between  mind  and 
nature.  This  is  an  experience  of  no  mean  moral  value,  to 
realize  that  our  mathematical  procedure  is  attuned  to  the 
harmony  of  the  universe.”  In  mathematics,  then,  the  human 
mind  relates  itself  to  the  Supreme  Intelligence,  whose  thought 
is  manifest  in  the  universe;  and  the  contact  must  leave  its 
imprint  on  our  lives.  This  leads  us  to  the  consideration  of  the 
religious  value  in  the  study  of  mathematics. 

Man  has  an  innate  desire  from  early  years  to  reach  towards 
the  Infinite  and  the  Eternal.  Clerk  Maxwell,  towards  the  end 
of  his  life,  said,  “I  have  looked  into  most  philosophical  systems, 
and  I  have  found  that  none  of  them  will  work  without  God.” 
Mathematics  reaches  farther  and  with  greater  certainty  than  any 
other  philosophy;  towards  a  Supreme  Being,  a  great  First  Cause. 
It  looks  down  the  vistas  of  the  ages  past;  and  into  the  dimness 
of  the  eons  to  come;  and  the  human  mind  is  awed  by  the  sub¬ 
lime  majesty  of  the  Divine.  Prof.  D.  E.  Smith  says,  “The  proper 
study  of  mathematics  gives  humanity  a  religious  sense  that 
cannot  be  fully  developed  without  it.  .  .  .  In  the  history  of 
the  world,  mathematics  had  its  genesis  in  the  yearning  of  the 
human  soul  to  solve  the  mystery  of  the  universe,  in  which  it 
is  a  mere  atom.  ...  It  seems  to  have  had  its  genesis  as  a 
science  in  the  minds  of  those  who  followed  the  course  of  the 
stars,  to  have  had  its  early  applications  in  relation  to  religious 
formalism,  and  to  have  its  first  real  development  in  the  effort 
to  grasp  the  Infinite.  And  even  today,  even  after  we  have  been 
pushing  back  the  sable  curtains  for  so  many  long  centuries — - 
even  today  it  is  the  search  into  the  Infinite  that  leads  us  on.” 
Col.  Parker  (“Pedagogics,”  p.  46)  says,  “I  can  assert  that, 
from  the  beginning,  man’s  growth  and  development  have  utterly 
depended,  without  variation  or  shadow  of  turning,  upon  his 
search  for  God’s  laws,  and  his  application  of  them  when  found, 
and  that  there  is  no  other  study  and  no  other  work  of  man. 
We  are  made  in  His  image,  and  through  the  knowledge  of  His 
laws  and  their  application  we  become  like  unto  Him,  we  ap¬ 
proach  that  image.”  .  .  .  “There  is  but  one  study  in  this  world 
of  ours,  and  I  call  it,  in  one  breath,  the  study  of  law,  and  the 
study  of  God.”  That  the  study  of  mathematics  is  the  study  of 


218 


SCHOOL  SCIENCE  AND  MATHEMATICS 


God’s  laws  and  so  must  lead  to  God  there  can  be  no  question. 
Plato,  “God  eternally  geometrizes.”  C.  J.  Keyser  says,  “Again 
it  is  in  the  mathematical  doctrine  of  invariance,  the  realm 
wherein  are  sought  and  found  configurations  and  types  of  being 
that,  amid  the  swirl  and  stress  of  countless  hosts  of  transforma¬ 
tions,  remain  immutable,  and  the  spirit  dwells  in  contempla¬ 
tion  of  the  serene  and  eternal  region  of  the  subtle  law  of  Form 
— it  is  there  that  Theology  may  find,  if  she  will,  the  clearest 
conceptions,  the  noblest  symbols,  the  most  inspiring  intimations, 
the  most  illuminating  illustrations,  and  the  surest  guarantees 
of  the  object  of  her  teaching  and  her  quest,  an  Eternal  Being, 
unchanging  in  the  midst  of  the  universal  flux.” 

“And  reason  now  through  number,  time,  and  space 
Darts  the  keen  lustre  of  her  serious  eye; 

And  learns  from  facts  compar’d  the  laws  to  trace 
Whose  long  procession  leads  to  Deity.” 

— Jas.  Beattie,  “The  Minstrel,”  Bk.  3. 

No  wonder  the  mind  is  fascinated  by  the  fields  opened  through 
the  study  of  mathematics.  One  is  led  to  a  spirit  of  reverence 
when  he  contemplates  the  human  intellect  as  revealed  in  the  ever 
unfolding  and  almost  limitless  range  of  mathematical  achieve¬ 
ment.  It  reveals  and  inspires  Godlikeness.  Mathematics 
deals  with  the  eternal  verities:  it  is,  if  you  will,  the  majestic 
mountain  peak  that  rises  in  sublimity  above  the  clouds  of  doubt 
and  uncertainty  and  basks  in  the  sunlight  of  eternal  truth. 

These  values  in  the  study  of  mathematics  will  suggest  the 
more  important  aims  and  purposes  for  its  study  and  its  teach¬ 
ing.  The  teacher  will  do  well  to  have  them  in  mind  as  a  back¬ 
ground  for  his  teaching;  in  this  way  they  will  permeate  and 
vitalize  his  work.  But  this  is  not  enough.  The  pupil  must  also 
appreciate  their  importance,  to  give  sufficient  motive  for  his 
work,  to  satisfy  him  that  the  study  is  worth  his  effort.  Not  all 
pupils  will  appreciate  these  to  the  same  extent.  The  teacher 
should  ascertain  the  interests  of  the  student  and  make  the 
appeal  chiefly  in  accordance  with  these  interests.  Bertrand 
Russell  says,  “Every  great  study  is  not  only  an  end  in  itself, 
but  also  a  means  of  creating  and  sustaining  a  lofty  habit  of  mind ; 
and  this  purpose  should  be  kept  always  in  view  throughout 
the  teaching  and  learning  of  mathematics.”  The  same  is  true 
of  all  purposes;  neither  teacher  nor  student  should  work  blindly 
without  knowing  what  to  expect  as  a  result  of  his  effort. 

We  have  made  our  defense  for  the  teaching  of  mathematics 
in  high  schools  when  we  have  shown  that  some  of  these  values 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


219 


can  be  realized  alone  by  the  study  of  mathematics,  and  that 
others  can  be  realized  better  through  mathematics  than  by  the 
study  of  any  other  subject.  We  have  shown  that  mathematics 
is  most  important  in  its  culture  values — that  it  is  indispensable 
to  everyone  who  would  live  his  best.  We  might  add  the  words 
of  President  Hadley  of  Yale,  “The  value  of  an  education  large¬ 
ly  consists  in  studying  facts  that  will  not  be  used  in  after  life, 
by  methods  that  will  be  used.”  Dr.  Snedden,  of  Teachers  Col¬ 
lege,  recognizes  the  fact  that  the  chief  claim  for  the  study  of 
mathematics  is  on  the  cultural  side.  In  “Problems  of  Secondary 
Education”  (recently  published),  p.  223,  he  says,  “I  am  con¬ 
vinced  that  the  prominence  of  algebra  (and  geometry)  in  secon¬ 
dary  education  rests  not  so  much  upon  faith  in  its  usefulness 
as  a  tool  of  further  learning  as  upon  belief  in  its  value  as  a 
means  of  ‘mental  training’  and  upon  the  faith  that  somehow 
a  knowledge  of  algebra  is  essential  to  general  culture.”  Again 
(p.  229),  speaking  of  a  secondary  school  curriculum,  he  would 
“Seek  to  develop  a  ‘culture’  course  in  mathematics  which  should 
prove  attractive  to  students  seeking  to  inform  themselves  about 
the  world  in  which  they  live;  this  to  include  some  account  of 
the  evolution  of  mathematics  as  a  human  tool  and  as  a  means 
of  interpretation,  as  well  as  a  survey  of  modern  applications 
of  mathematics  to  the  understanding  of  the  universe  and  to 
the  work  of  the  world.  Just  as  many  of  us  can  respond  to  operas, 
epics,  and  great  paintings  without  being  artists  in  these  fields, 
so  I  think  many  could  be  led  to  appreciate  the  place  of  mathe¬ 
matics  without  becoming  mathematicians.”  In  the  light  of 
what  we  have  said,  we  would  change  “should  prove  attractive,” 
to  “should  be  made  attractive,”  throwing  responsibility  upon 
the  teacher.  We  would  require  a  course  in  mathematics  of 
everyone.  We  would  make  this  course  to  include  the  actual 
study  of  mathematics;  for  while  we  would  not  seek  to  make 
mathematicians  of  all  students,  in  the  sense  of  making  each  a 
specialist,  everyone  needs  to  study  mathematics  as  well  as  to 
study  about  it.  And  further,  the  cultural  value  of  mathematics 
is  of  immeasurably  greater  importance  than  the  mere  getting 
of  information  about  things.  However,  a  knowledge  of  mathe¬ 
matics  is  essential  in  much  of  life’s  work;  and  it  is  evident  that 
it  will  become  more  and  more  important  as  a  means  of  investiga¬ 
tion  in  practically  every  field  of  endeavor. 

Let  us  remember,  withal,  that  the  greatest  values  and  aims 
in  the  study  of  any  subject,  like  all  the  greatest  things  in  life, 


220 


SCHOOL  SCIENCE  AND  MATHEMATICS 


cannot  be  easily  reduced  to  exact  measurement.  We  have 
heard  Dr.  Smith  say  that  he  wished  someone  might  measure 
the  value  of  the  early  study  of  Greek  in  his  life.  So  far  no  one 
has,  and  it  is  not  likely  that  the  psychologist  ever  will  invent 
a  measure  for  such  values.  Therefore,  while  we  satisfy  ourselves 
and  seek  to  satisfy  others  by  giving  reasons  for  our  faith  in,  and 
enthusiasm  for,  the  subject  we  teach,  we  do  not  hope  to  close 
the  door  to  the  questions  of  doubters  and  critics.  Nor  do  we 
think  the  door  can  ever  be  closed  or  ever  should  be.  It  is  the 
glory  of  mathematics  that,  while  it  has  had  destructive  critics 
through  the  centuries,  it  has  survived  them  all  and  is  marching 
steadily  onward  to  a  higher  and  a  surer  place  in  our  civiliza¬ 
tion.  Like  everything  worth  while,  it  has  its  critics  and  enemies, 
but  these  ultimately  contribute  to  its  strength.  They  aid 
mathematics  in  leading  its  advocates  to  establish  more  firmly 
its  claims,  to  adjust  it  better  to  changed  conditions;  and  they 
are  the  means  of  heralding  its  worth  to  the  multitude.  This  is 
the  logical  outcome  of  present  criticism. 

(To  be  Continued) 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


313 


VALID  AIMS  AND  PURPOSES  FOR  THE  STUDY  OF 
MATHEMATICS  IN  SECONDARY  SCHOOLS. 

By  Alfred  Davis, 

Francis  W.  Parker  School, 

330  Webster  Ave.,  Chicago. 

Chairman  of  a  Committee  of  the  Mathematics  Club  of  Chicago 
Appointed  to  Investigate  This  Topic. 

(Continued  from  March  number) 

Part  III. 

If  we  are  to  realize  our  aims  and  purposes  in  the  teaching  of 
mathematics  it  may  be  necessary,  as  times  and  conditions 
change,  to  vary  both  subject  matter  and  methods  of  teaching 
it.  However,  such  change  is  evidence  of  life,  not  of  death. 

The  New  England  Report,  already  referred  to,  says,  “Some 
teachers  of  mathematics  in  the  high  schools  are  inadequately 
prepared  for  their  work,  perhaps  teaching  it  only  incidentally. 
Conditions  in  normal  and  grammer  schools  are  probably  worse.” 
Matthew  Arnold  says,  “The  plan  of  employing  teachers  whose 
attainments  do  not  rise  far  above  the  level  of  the  attainments 
of  their  scholars  has  been  tried  and  it  has  failed.”  By  far  too 
much*  emphasis  is  put  on  methods  in  schools  of  education  for 
prospective  teachers  when  these  would-be  teachers  have  not 
thoroughly  mastered  the  subjects  they  seek  to  teach.  Not  that 
any  teacher  can  know  too  well  how  to  teach,  but  the  study 
about  education  should  never  be  at  the  expense  of  education  itself. 
The  scholarship  requirements  for  teachers  should  be  higher  than 
they  are;  indeed,  we  can  scarcely  make  them  too  high;  for  superior 
scholarship  in  the  subject  one  is  to  teach  is  of  supreme  impor¬ 
tance,  and  all  of  us  would  be  happier  with  greater  efficiency  in 
our  chosen  field.  Much  of  the  criticism  of  mathematics  today 
is  from  those  who  are  perfectly  well  informed  as  to  how  it  ought 
to  be  taught,  but  they  have  little  knowledge — and  less  apprecia¬ 
tion — of  the  subject.  Prof.  L.  T.  More  ( Nation ,  May  3, 
1917)  says,  “A few  months  in  a  schoolroom  acting  as  an  assistant 
to  an  experienced  teacher  after  a  sound  college  course  will 
prepare  a  person  for  teaching  more  effectively  than  pedagogical 
courses.”  The  plan  is  worth  thinking  about. 

On  the  other  hand,  granting  adequate  scholarship,  all  who 
enter  the  teaching  profession  need  technical  training  in  teaching, 
more  than  most  of  them  get  at  present.  Knowledge  of  a  sub¬ 
ject  does  not  necessarily  imply  ability  to  teach  it;  it  is  frequently 
assumed  that  it  does,  but  some  brilliant  students  have  made 


314 


SCHOOL  SCIENCE  AND  MATHEMATICS 


very  poor  teachers.  It  is  a  crime  to  make  the  classroom  a  labora¬ 
tory  in  which  the  would-be  teacher  shall  try  by  hit-and-miss 
efforts  to  gain  experience.  Doctors  of  medicine  do  not  gain 
all  their  preparation  by  experimenting  on  the  helpless  sick. 
Lawyers  are  not  entrusted  with  our  reputations  before  they 
have  earned  to  practice  law.  The  teacher  in  training  should  be 
under  the  direction  of  an  expert.  Our  material,  the  plastic 
minds  of  boys  and  girls,  is  too  valuable  to  be  trifled  with.  Be¬ 
sides,  we  have  too  much  at  stake  in  the  reputation  of  our  subject 
to  take  such  chances.  The  teacher  must  learn  to  see  the  sub¬ 
ject  from  the  pupil’s  angle  and  to  present  it  effectively;  to  accom¬ 
plish  this  he  must,  while  practicing  under  an  expert,  master 
the  fundamentals  of  psychology  and  of  pedagogy.  Above  all, 
the  teacher’s  preparation  should  draw  out  his  individuality, 
the  thing  that  makes  him  different  from  every  other  teacher; 
and  it  should  enable  him  to  avoid  the  ruts  from  the  beginning. 

Much  of  the  inability  of  students  to  master  mathematics 
is  due  to  poor  teaching  at  some  point  in  their  career.  Those 
who  cannot  master  a  reasonable  amount  of  mathematics  are 
probably  no  more  numerous  than  physical  defectives.  Someone 
has  said  that  algebra  is  “fool  proof,”  and  this  has  been  assumed 
true  by  many  principals  in  assigning  work  to  teachers.  Dr. 
Snedden,  (“Problems  of  Secondary  Education,”  p.  225)  says, 
“Algebra  is  one  of  the  easiest  of  secondary  school  subjects  to 
teach  with  a  certain  degree  of  apparent  effectiveness.  Lessons 
and  hard  tasks  can  be  assigned  easily,  and  a  very  duffer  of  a 
teacher  can  make  pupils  work  slavishly  on  this  subject.  In 
most  small  high  schools  today  it  will  be  found  that  the  teacher 
with  least  special  preparation  for  his  work  is  usually  teaching 
algebra.  The  case  here  is  somewhat  analogous  to  the  practice 
of  ‘electric’  and  ‘magnetic’  healing  by  ‘near’  physicians.”  This 
is  a  somewhat  frank  statement  that  the  trouble  with  secondary 
mathematics  is  not  with  the  subject  but  with  the  teaching  of 
it.  Poor  teaching  gives  a  subject  the  reputation  of  being  diffi¬ 
cult.  When  pupils  pass  the  word  along  that  algebra  or  geometry 
is  hard  and  uninteresting  the  battle  is  lost  before  it  is  begun. 
Even  a  good  teacher  will  have  difficulty  in  overcoming  this 
prejudice. 

Another  source  of  trouble  is  the  assumption  that  when  a 
pupil  masters  a  principle  this  guarantees  his  ability  to  use  it. 
Such  is  not  the  case.  Many  of  our  best  students  appear  sadly 
inefficient  when  they  leave  high  school.  Application  is  quite 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


315 


as  important  a  part  of  our  work  as  the  teaching  of  principles. 
Of  what  value  is  a  tool  unless  we  are  taught  the  actual  use  of 
it?  Prof.  Judd  (“The  Psychology  of  High  School  Subjects/’  p. 
130)  says,  “It  is  contrary  to  experience  to  assume  that  students 
can  apply  mathematics  to  the  other  sciences  or  to  the  practical 
affairs  of  life  unless  they  are  trained  to  see  mathematical  rela¬ 
tions  in  other  forms  than  those  in  which  they  are  commonly 
presented  in  the  schools.  The  student  who  knows  the  abstract 
demonstrations  of  geometry,  but  does  not  realize  that  knowledge 
of  space  is  involved  in  every  manufacturing  operation,  in  every 
adjustment  of  agriculture  and  practical  mechanics,  is  only  half 
trained.  Application  must  be  a  phase,  and  an  explicit  phase, 
of  school  work.  Application  is  as  different  from  pure  science 
as  pure  sciences  are  different  from  each  other.” 

In  addition  to  the  inefficiency  of  teachers  we  have  the  ineffi¬ 
ciency  of  students  as  such.  This  is  probably  due  to  two  causes. 
First,  the  idea  is  somewhat  prevalent  in  the  elementary  schools 
that  a  pupil  may  gain  an  education  by  being  amused.  He  is 
not  trained  to  feel  nor  to  know  personal  responsibility.  His 
education  is  entirely  a  matter  for  the  teacher  to  look  after.  He 
even  resents  any  attempt  to  secure  his  initiative  if  the  effort 
he  is  to  make  savors  of  drudgery  or  requires  hard  thinking. 
School  work  must  be  a  pleasure  and  without  responsibility. 
^Second,  there  is  a  tendency  on  the  part  of  college  students  to 
ignore  study.  This  spirit  finds  its  way  into  the  high  school, 
which  so  often  tries  to  imitate  the  college.  Consequently, 
many  who  could  be  good  students  fail;  mathematics,  requiring 
the  sort  of  effort  they  know  nothing  about,  is  impossible  with 
them.  Ability  in  mathematics  may,  usually,  be  considered 
synonymous  with  ability  to  be  a  good  student.  Prof.  Schultze 
(“Teaching  of  Mathematics  in  Secondary  Schools,”  p.  25) 
says,  “It  is  a  common  experience  to  see  a  pupil  in  the  upper 
grades  suddenly  wake  up  to  the  meaning  of  mathematics  and 
thereby  change  his  attitude  towards  study  in  general.”  A  good 
standing  in  high  school  is  almost  certain  to  guarantee  a  good 
college  record  and  a  successful  life.  This  matter  has  been 
splendidly  treated  by  Wm.  T.  Foster,  President  of  Reed  College, 
Portland,  Oregon,  in  “Should  Students  Study”  (Harper’s  Month¬ 
ly,  September,  1916).  He  raises  the  question,  “Are  good  students 
in  high  school  more  likely  than  others  to  become  good  students 
in  college?”  Three  colleges  in  as  many  states  are  considered. 
Of  hundreds  of  students  in  the  University  of  Wisconsin,  above 


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SCHOOL  SCIENCE  AND  MATHEMATICS 


80  per  cent  of  those  in  the  first  quarter  in  the  high  school  re¬ 
mained  in  the  upper  half  of  their  classes  throughout  the  four 
years  of  their  university  course,  and  above  80  per  cent  of  those 
who  were  in  the  lowest  quarter  in  high  school  did  not  rise  above 
mediocre  scholarship  in  the  university.  Only  one  in  five  hun¬ 
dred  of  those  in  the  lowest  quarter  reached  highest  rank  in 
the  university.  The  University  of  Chicago  found  that  students 
that  failed  to  receive  in  high  school  an  average  higher  than 
p£ss  by  at  least  one-fourth  of  the  difference  between  the  pass¬ 
ing  mark  and  100  per  cent  failed  in  college;  such  students  are, 
therefore,  not  admitted;  where  exceptions  are  made  the  record 
in  college  is  seldom  satisfactory.  Reed  College,  at  its  fall 
opening  five  years  ago,  admitted  only  those  students  who  ranked 
in  the  first  third  in  the  preparatory  schools:  about  20  per  cent 
were  exceptions  to  this  rule  and  2  per  cent  were  below  the 
median  line;  these  exceptions  were  selected  as  the  best  below 
the  first  third.  Of  these  exceptions,  practically  none  rose  above 
the  lowest  quarter  in  their  college  classes.  The  same  results 
are  shown  to  be  true  of  those  who  go  from  college  to  the  profes¬ 
sional  schools.  Surely  “promise  in  high  school  becomes  per¬ 
formance  in  college/’  and  the  mediocre  in  high  school  are  out 
of  the  race.  President  Foster  says,  “If  all  these  studies  prove 
anything,  they  prove  that  there  is  a  long  chain  of  causal  con¬ 
nections  binding  together  the  achievements  of  a  man’s  life  and 
explaining  the  success  of  a  given  moment.  .  .  .  Luck  is  about  as 
likely  to  strike  a  man  as  lightning  and  about  as  likely  to  do 
him  any  good.  The  best  luck  a  young  man  can  have  is  the  firm 
conviction  that  there  is  no  such  thing  as  luck  and  that  he  will 
gain  in  life  just  about  what  he  deserves  and  nor  more.  .  .  . 
At  a  convention  of  teachers  not  long  ago  a  speaker  ridiculed  a 
German  boy  who,  upon  failing  in  a  recitation,  put  his  head 
upon  his  desk  and  cried.  He  said  he  had  never  seen  such  a 
boy  in  the  schools  of  this  country.  .  .  .  Nothing  seems  to 
promise  failure  in  the  tasks  of  tomorrow  with  greater  certainty 
than  failure  in  the  studies  of  today.  .  .  .  Among  teachers  the 
greatest  number  of  criminals  are  not  those  who  kill  their  young 
charges  with  overwork,  but  those  who  allow  them  to  form 
the  habit  of  being  satisfied  with  less  than  the  very  best  there  is 
in  them.” 

From  a  study  by  President  Lowell,  of  Harvard  ( Educational 
Review,  October,  1911),  it  appears  that  President  Foster’s 
conclusions  apply  to  Harvard  students.  Of  609  who  graduated 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


317 


from  college  with  A.  B.  plain,  only  6.6  per  cent  obtained  cum 
laudi  in  Law  School.  Students  of  mathematics  attained  high¬ 
est  honors  in  Law  School.  Students  of  classics  stood  next. 
The  qualities  of  diligence,  perseverance,  and  intensity  of  appli¬ 
cation  acquired  in  the  study  of  mathematics  secured  a  higher 
degree  of  success  than  was  obtained  by  others.  The  pupil 
should  be  impressed  with  the  importance  of  proper  habits  of 
study  and  he  should  be  aided  in  forming  these  habits.  If  he 
finds  the  work  very  hard,  let  him  profit  by  the  determination  of 
Robert  Bruce  as  he  watched  the  spider,  after  many  futile  efforts, 
finally  reach  the  ceiling.  If  he  is  inclined  to  waste  time  let  him 
get  inspiration  from  the  words  of  Hotspur  before  the  battle  of 
Shrewsbury: 

“Oh,  gentlemen,  the  time  of  life  is  short; 

To  spend  that  shortness  basely  were  too  long, 

If  life  did  ride  upon  a  dial’s  point, 

Still  ending  at  the  arrival  on  an  hour.” 

The  New  England  Report  says,  “There  is  real  danger  that 
depreciation  of  mathematics  by  persons  supposed  to  be  experts 
in  matters  of  secondary  education  may,  unless  vigorously  met, 
exert  an  unfavorable  and  undue  influence  on  public  opinion.” 
It  would  seem  desirable  that  teachers  of  mathematics  assert 
their  claims  more  energetically,  for  it  is  not  always  safe  to  assume 
that  right  will  win  without  an  advocate.  It  is  not  necessary  to 
enumerate  these  criticisms;  we  are  all  familiar  with  them.  Some 
have  been  justified  and  have  already  resulted  in  improved 
courses  and  methods  in  the  better  high  schools;  for  no  subject 
is  so  perfect  as  to  need  no  improvement;  and  any  subject  will 
soon  be  out  of  date  if  it  lacks  the  elasticity  which  enables  it  to 
fit  new  conditions.  There  is  still  room  for  much  improvement. 
Dr.  D.  E.  Smith  says,  “I  think  unquestionably  there  has  been  too 
high  a  pace  set  in  the  matter  of  abstract  drudgery.  I  think 
there  is  no  question  but  that 'we  must  harmonize  and  vitalize 
our  algebra.”  Other  criticisms  have  been  answered  time  and 
time  again,  but  like  Banquo’s  ghost  they  will  not  down.  Prof. 
R.  D.  Carmichael,  in  Science ,  May  18,  1917,  says,  “It  seems 
that  no  body  of  thought  has  been  of  more  importance  in  human 
progress  and  at  the  same  time  been  criticized  more  freely  than 
the  science  of  mathematics.  Much  of  this  criticism  seems  to 
be  good  natured  and  to  amount  to  little  more  than  a  quasi  - 
humorous  way  of  expressing  the  critic’s  unashamed  ignorance. 
At  first  sight  one  might  treat  this  as  harmless;  but  from  the 


318 


SCHOOL  SCIENCE  AND  MATHEMATICS 


point  of  view  of  general  interest  it  can  hardly  be  passed  over  in  such 
a  way.  How  this  ignorance  is  to  be  overcome  I  cannot  say. 
Perhaps  one  of  the  first  requisites  is  to  find  some  means  of  over¬ 
coming  the  shamelessness  with  which  individuals  otherwise 
well  trained  contemplate  their  own  ignorance  of  mathematics.” 
Prof.  Chas.  N.  Moore,  University  of  Cincinnati,  in  The  Ameri¬ 
can  Mathematical  Monthly ,  February,  1916,  says,  “The  move¬ 
ment  against  mathematics  is,  for  the  most  part,  confined  to  a 
group  of  theorists  who  feel  that  they  must  advocate  something  new 
in  order  to  convince  their  readers  that  they  are  investigators. 
This  group,  however,  has  made  up  in  volume  of  sound  what  it 
has  lacked  in  numbers.  .  .  .  The  statement  that  there  is  at  the 
present  time  much  uncertainty  as  to  the  educational  value  of 
algebra  and  geometry  will  not  bear  examination.”  Dr.  D.  E.  Smith, 
in  T.  C.  Record ,  May,  1917,  says,  “It  is  by  no  means  the  ad¬ 
vanced  educator  who  denies  a  disciplinary  value  to  geometry; 
it  is  rather,  I  think,  either  the  educator  who  is  slipping  behind 
in  the  race,  or  the  one  who  has  never  been  in  the  race  at  all. 
If  you  do  not  think  so,  ask  a  man  like  Professor  Thorndike. 
If  anyone  says  to  me  that  we  have  statistics  to  show  that  young 
people  spend  a  year  on  a  subject  whose  chief  purpose  is  the  logical 
proving  of  statements  and  are  not  thereby  made  more  logical 
in  their  other  lines  of  mental  activity,  now  or  in  years  to  come, 
then  I  tell  him  frankly  that  not  only  do  I  not  believe  it  but 
that  scientific  men  generally  do  not.”  Our  critics  seem  to  be 
so  fascinated  by  the  weight  and  worth  of  their  own  ideas  that 
no  reply  can  reach  them.  Some  seek  to  surprise  us  with  the 
resurrection  of  issues  that  have  been  dead  for  ages.  Others, 
in  the  face  of  irrefutable  argument,  do  the  seemingly  impossi¬ 
ble;  they  absolutely  ignore  the  other  side  of  the  issue;  they 
continue  their  argument  as  though  it  were  impossible  for  them 
to  be  mistaken  or  for  anyone  to  make  a  reply  worthy  of  notice. 
Prof.  Paul  Shorey,  in  “The  Assault  on  Humanism”  {The  Alantic 
Monthly,  June,  1917),  says,  “They  either  have  not  read  the 
literature  which  they  controvert,  or  they  intentionally  ignore 
it.  They  do  not  inform  their  readers  of  its  existence,  and  they 
do  not  even  tacitly  amend  their  own  arguments  to  meet  its 
specific  contentions.  In  controversy,  this  is  what  Lincoln 
.called  Tush  whacking.’  In  the  authors  of  the  textbooks  of  the 
science  or  the  history  of  education  it  is  the  abandonment  of 
the  scientific  for  the  frankly  partisan  attitude.  .  .  .  They  not 
only  argue  as  partisans  against  the  classics  but  they  systematical- 


1 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


319 


ly  suppress  both  the  arguments  and  the  bibliography  of  the  case 
2  for  the  classics.  .  .  .  The  principal  effort  of  the  classicist  who 

aims  at  argument  rather  than  eloquence  must  be  to  shame  his 
opponents  from  their  unfair  tactics,  their  neglect  of  the  evidence, 
their  preposterous  logic,  and  to  urge  the  educated  public  to 
examine  the  matter  for  themselves.  He  must  wearily  repeat 
his  old  list  of  ‘must  nots’  and  ‘don’ts.’  ”  These  statements  apply 
with  equal  force  to  many  of  the  critics  of  mathematics. 

As  teachers  of  mathematics,  we  are  too  often  the  victims  of 
men  officially  higher  up,  who  know  little  of  mathematics  or  of 
its  teaching,  and  whom  we  allow  to  do  our  talking  and  planning 
for  us.  Why  should  courses  and  curricula  in  mathematics  be 
arranged  and  criticized  by  principals  and  superintendents  who 
have  no  appreciation  of  the  subject?  It  seems  to  be  different 
in  Europe.  Prof.  J.  C.  Brown,  in  a  recent  bulletin  of  the  U.  S. 
Bureau  of  Education,  says,  “European  school  men  believe  that 
a  course  in  mathematics  should  be  planned  by  those  who  know 
mathematics  rather  than  by  educators  who  are  practically  ignor¬ 
ant  of  the  subject^  Educators  not  familiar  with  mathematics 
may  aid  in  an  advisory  way;  but,  if  the  subject  does  what  we 
*  claim  for  the  student,  it  must  have  done  that  in  some  measure 
for  the  teacher;  who  then  can  be  so  well  fitted  to  mould  the 
subject  into  better  form  or  to  adjust  its  courses? 

Some  of  the  difficulties,  and  lack  in  interest  of  students  in 
the  study  of  algebra  and  geometry,  are  the  result  of  preconceived 
notions,  the  echoes  of  what  educational  theorists  have  said. 
We  have  heard  a  teacher  say  that  it  is  a  pity  that  the  active 
productive  minds  of  young  people  should  be  burdened  with  so 
formal  a  study  as  algebra.  A  student  who  is  looking  for  an  easy 
time  and  who  finds  algebra  hard  work  thinks  he  has  sufficient 
authority  in  such  a  statement  for  loafing  on  the  subject.  This 
point  is  cleverly  made  by  Prof.  Crathorne  in  a  parable  given  in 
“Algebra  from  the  Utilitarian  Standpoint”  (School  Science 
and  Mathematics,  vol.  16,  pp.  418-431).  After  considerable 
discussion  on  the  part  of  Dr.  Highbrow,  Dr.  Practical  Man,  Dr. 
Brown,  and  an  Average  Parent  regarding  difficulties  in  algebra 
and  dislike  for  the  subject,  which  the  Average  High  School 
Pupil  seemed  to  be  experiencing,  the  pupil  broke  in  with,  “But 
I  don’t  dislike  algebra.  I  never  did;  but  so  many  people  argued 
that  I  disliked  it  that  I  began  to  think  there  was  something  wrong 
with  it.”  That  much  of  the  lack  of  interest  on  the  part  of 
students  towards  mathematics  is  merely  assumed  is  shown 


320 


SCHOOL  SCIENCE  AND  MATHEMATICS 


in  a  report  by  this  club  published  in  School  Science  and 
Mathematics,  October,  1916.  A  test  was  given  in  three  high 
schools — University  High  and  Hyde  Park,  Chicago,  and  Oak  Park, 
Ill. — to  determine  from  the  statements  of  students  in  these  schools 
the  measure  of  enjoyment  they  found  in  their  various  studies. 
Out  of  a  total  of  2,018  pupils  reported  as  taking  mathematics, 
1,769,  or  87.7  per  cent,  said  they  got  some  enjoyment  from  the 
study;  991,  or  49.1  per  cent,  claimed  to  get  very  much  enjoy¬ 
ment;  and  only  229,  or  12.3  per  cent,  said  they  got  no  enjoy¬ 
ment.  With  one  year  of  mathematics  (at  least)  required  of 
all  students  in  these  schools,  and  in  view  of  present  criticisms, 
these  results  are  remarkable.  Subjects  which  ranked  higher 
were  purely  elective.  It  is  the  opinion  of  Prof.  J.  W.  Johnson 
that  49  per  cent  of  the  pupils  who  study  mathematics,  includ¬ 
ing  the  high  schools  of  the  entire  country,  would  vote  that  they 
got  very  much  enjoyment  from  the  study  of  mathematics. 

It  would  be  a  mistake  to  try  to  persuade  ourselves  that 
no  criticisms  are  worth  our  attention.  The  destructive  critic 
always  defeats  his  own  ends,  and  is  unworthy  of  serious  consid¬ 
eration;  but  the  constructive  critic  is  always  welcome;  providing 
he  maintains  a  friendly  spirit.  We  must  have  the  assistance 
of  the  latter.  G.  St.  L.  Carson,  in  “Mathematical  Education,” 
pp.  61  and  85,  says,  “The  whole  world  is  going  through  a  trans¬ 
formation,  due  in  part  to  scientific  and  mechanical  invention 
and  in  part  to  the  growth  of  separate  nations,  each  with  its  own 
methods  and  ideals,  of  which  no  man  can  see  the  outcome. 
Our  function,  the  function  of  all  teachers,  is  to  produce  men 
and  women  competent  to  appreciate  these  changes  and  to  take 
their  part  in  guiding  them  so  far  as  may  be  possible.  Mathe¬ 
matical  thought  is  one  fundamental  equipment  for  this  pur¬ 
pose,  but  mathematical  teaching  has  not  hitherto  been  devoted 
to  it  because  the  need  has  but  recently  arisen.  But  now  that  it 
has  arisen  and  is  appreciated,  we  must  meet  it  or  sink,  and  sink 
deservedly.  Neither  the  arid  formalism  of  older  days  nor  the 
workshop  reckoning  introduced  of  late  will  save  us.  The  only 
hope  lies  in  grasping  that  inner  spirit  of  mathematics  which 
has  in  recent  years  simplified  and  coordinated  the  whole  struc¬ 
ture  of  mathematical  thought,  and  in  relating  this  spirit  to  the 
complex  entities  and  laws  of  modern  civilization.  .  .  .  Salva¬ 
tion  from  our  present  difficulties  can  come  only  from  the  efforts 
and  experiments  of  teachers  themselves.  Educational  matters 
are  in  a  ferment.  Men  are  asking  more  and  more  insistently 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


321 


why  this  and  that  are  done,  and  they  are  right  in  their  insistence. 
Unless  fitting  answers  are  ready,  our  work  will  stand  condemned; 
the  degradation  of  our  subject  to  the  domain  of  purely  immedi¬ 
ate  utility  will  surely  follow,  as  also  the  loss  of  that  higher 
mental  training  which  is  so  essential  to  the  formation  of  an 
efficient  citizen.”  To  meet  this  situation  we  would  make  the 
following  suggestion: 

Mathematics  should  be  required  of  every  secondary  school 
pupil.  This  required  work  may  consist  of  courses  alloted  to 
the  junior  high  school,  if  that  institution  is  to  prevail;  or  it  may, 
under  the  present  arrangement,  be  reduced  to  one  year  and  con¬ 
sist  of  algebra  and  geometry,  with  some  reference  to  the  use  of 
the  trigonometric  functions.  The  purpose  being,  in  any  case, 
to  give  to  all,  early  in  their  course,  some  command  of  mathe¬ 
matics  as  a  tool;  to  give  some  knowledge  of  its  historical  develop¬ 
ment;  to  instill  an  appreciation  of  the  importance,  possibilities, 
and  beauties  of  the  subject;  and,  above  all,  to  give  the  pupil  a 
chance  to  find  himself.  Those  who  have  little  aptitude  for  mathe¬ 
matics  would  not  elect  additional  courses,  and  yet  would  realize 
some  of  the  benefits  from  its  study.  This  would  probably 
reduce  pupil  mortality  in  our  h  gh  schools  and  remove  much 
of  the  grounds  for  present  criticism.  On  the  other  hand,  such 
an  arrangement  would  throw  responsibility  on  the  teacher  to 
interest  his  pupils  in  mathematics  while  taking  the  required 
courses  so  that  they  would  elect  other  courses  in  the  subject. 
This  might  aid  in  securing  better  teaching  and  better  teachers. 
This  club  has  already  expressed  its  approval  of  reducing  the 
required  mathematics  in  high  school  to  one  year.  Prof.  Paul 
H.  Hanus,  of  Harvard,  in  his  book,  “A  Modern  School,”  pp. 
76  and  83,  says,  “Permanent  lack  of  interest  in  a  given  field 
of  work  is  an  indication  of  corresponding  incapacity;  for  growing 
interest  and  capacity  always  go  together.  ...  No  pupil  should 
be  required  to  pursue  a  study  after  it  is  clear  that  it  does  not 
appeal  to  him.  Under  most  circumstances  one  year  is  enough — 
and  it  is  not  too  much — to  ascertain  whether  a  study  does, 
or  does  not,  challenge  a  youth’s  interest  and  capacity.  .  .  .  vOne 
year  of  elementary  algebra  and  geometry  may  open  the  pupil’s 
mind  to  one  of  the  most  useful,  the  most  profound,  and  to  some 
minds  most  fascinating  systems  of  thought  which  man  has  de¬ 
veloped — a  result  which  can  never  be  expected  to  follow  from 
what  the  pupil  has  learned  in  the  narrow  field  covered  by  arith¬ 
metic.”  Regarding  the  nature  of  elementary  courses  in  mathe- 


322 


SCHOOL  SCIENCE  AND  MATHEMATICS 


matics,  Prof.  G.  A.  Miller,  in  the  preface  to  his  “Historical 
Introducton  to  Mathematics”  (1916),  says,  “Early  mathematical 
courses  should  be  more  informational,  especially  along  historical 
lines,  on  the  ground  that  knowledge  begets  interest  in  knowl¬ 
edge.”  That  a  well-planned  and  interesting  course  should  be 
required  there  can  be  no  question.  No  one  can  know  whether 
or  not  he  has  aptitude  for  the  subject  until  he  knows  some¬ 
thing  about  it.  One  cannot  know  anything  about  mathematics, 
to  such  an  end,  until  one  has  studied  the  subject. 

The  Lincoln  School,  established  this  fall  as  a  part  of  Teach¬ 
ers  College,  is  an  effort  to  determine  the  value  of  some  of  the 
modern  theories  of  education,  and  to  work  out  a  better  program 
of  study  than  we  have  at  present.  We  shall  await  with  interest 
and  shall  welcome  the  results  of  this  effort.  It  announces  that 
it  will  attempt  “in  the  subject  of  mathematics  to  develop  a 
course  which  connects  the  study  of  mathematics  with  its  use, 
adequate  provision  being  made  for  those  whose  special  abilities 
or  future  interests  relate  to  mathematics.  And  in  all  subjects, 
whenever  feasible,  effort  will  be  made  to  base  schoolwork  upon 
real  situations,  to  the  end  that  schoolwork  may  not  only  seem 
real  to  the  pupil,  but  be  so.”  It  would  seem  that  our  work  in 
algebra  and  geometry  might  be  more  completely  related  to  the 
interests  and  activities  of  the  pupil  than  at  present.  It  would 
seem  that  this  result  might  be  accomplished  by  a  greater  num¬ 
ber  and  variety  of  practical  problems,  even  though  these  had 
little  direct  relation  to  the  utility  of  the  subject  in  later  life. 
To  say  that  only  those  problems  which  are  directly  related  to 
the  doing  of  the  world’s  work  are  legitimate  is  to  place  mathe¬ 
matics  on  a  utilitarian  basis,  and  this  we  have  shown  to  be 
wrong.  The  values  we  have  suggested  can  be  realized  when 
the  pupil  is  thoroughly  interested  in  his  work  regardless  of  its 
actual  use  in  later  life.  However,  as  often  as  possible  problems 
should  be  useful  as  well  as  interesting.  Considerable  progress 
is  being  made  in  this  line  in  various  parts  of  the  country. 

Surely  all  important  departures  from  the  established  order 
of  things  should  be  made  slowly,  and  only  after  these  have  been 
carefully  tested.  Experiments  with  high  school  mathematics 
should  be  made  in  schools  adapted  to  that  use,  and  by  those 
who  are  expert  in  mathematics  as  well  as  in  the  teaching  of  the 
subject.  This,  of  course,  would  not  preclude  the  possibility 
of  important  discoveries  and  contributions  by  any  teacher 
situated  in  any  school. 


MATHEMATICS  IN  SECONDARY  SCHOOLS 


323 


■  The  high  school  is  not  the  place  to  specialize.  The  required 

■  courses  in  mathematics  must  be  adapted  at  the  same  time  to 
Bthe  pupil  who  does  not  and  to  the  one  who  does  go  to  college. 

■  Diversified  courses  may  have  a  place  as  electives  and  in  voca- 
B  tional  and  technical  schools.  Only  a  very  small  number  of  high 
I  school  pupils  can  have  any  adequate  notion  of  what  their  life’s 
I  work  should  be.  It  is  a  great  mistake  to  force — or  even  to  per- 
I  mit — them  to  follow  courses  that  may  exclude  them  from  the 
1  vocation  for  which  they  are  best  fitted.  President  Hadley, 

of  Yale,  says  that  only  about  8  per  cent  of  pupils  previous  to 
seventeen  years  indicate  inclinations  towards  any  vocation. 
Dr.  Joseph  Rausohoff,  an  eminent  surgeon  of  Cincinnati  says, 
(School  and  Society,  June  19,1915)  “Up  to  the  fifteenth  or  sixteenth 
year  the  average  boy  who  goes  to  a  high  school  can  have  no  idea 
as  to  the  work  he  expects  to  follow  in  later  life.  A  general 
course  will  give  the  boy  a  general  knowledge  which  will  later 
permit  him  to  develop  along  certain  lines,  as  his  bent  or  neces¬ 
sity  may  indicate.  Such  a  course  makes  the  possibility,  at  least, 
of  a  general  culture  which  will  permit  him  to  indulge  in  one  or 
another  intellectual  hobby  later  in  life.  I  would  above  all 
things  not  exclude  mathematics,  but  make  it  compulsory  in 
every  high  school  curriculum,  because  it  is,  after  all,  the  only 
study  which  will  inculcate  into  the  young  mind  that  absolute- 
precision  is  among  the  human  possibilities.”  Opportunity  for 
the  broadest  possible  education  must  be  given  to  everyone, 
i  This  is  an  essential  thing  in  a  democracy  where  equal  opportunity 
should  be  enjoyed  by  all.  We  cannot  say  to  one  pupil,  “The 
door  to  the  highest  attainments  in  education  is  open  to  you”; 
and  even  suggest  to  another,  by  limiting  his  field,  that  the 
same  door  is  practically  closed  to  him.  This  should  be  our 
ideal  in  curriculum  making  and  in  teaching,  and  it  should  be 
kept  constantly  before  both  teacher  and  student.  The  way  to 
the  highest  achievement  must  not  be  closed  to  anyone. 

We  have  quoted  freely  in  defense  of  our  arguments,  for  the 
words  of  a  thinker  carry  their  own  weight.  We  are  reminded 
of  the  words  of  Emerson,  in  “Letters  and  Social  Aims,  Quotation 
and  Originality”:  “A  great  man  quotes  bravely,  and  will  not 
draw  on  his  invention  when  his  memory  serves  him  with  a 
word  as  good.”  The  value  of  a  well-expressed  thought  or  of  a 
good  piece  of  work  depends  on  the  use  made  of  it — the  passing 
of  it  along.  We  have  quoted  from  specialists  in  mathematics, 
for  who  but  those  who  know  and  appreciate  a  subject  can  speak 


324 


SCHOOL  SCIENCE  AND  MATE 


cs 


with  authority  about  it.  Lest  this  should  seem  a  biased  attitude,  | 
we  have  also  quoted  specialists  in  other  lines,  and  have  e*en 
given  the  opinion  of  students  in  high  school.  Some  may  object 
that  mathematics  has  changed  since  some  of  these  statements 
were  made  concerning  it,  and  that  they  are  therefore  of 
doubtful  value.  While  mathematics  is  not  today  what  it  was 
yesterday,  it  is  not  less  but  more  important. 

That  mathematics  is  advancing  today  as  never  before  is  shown 
by  the  following  quotations: 

G.  A.  Miller,  “Historical  Introduction  to  Mathematical 
Literature,”  p.  22,  “It  would  be  very  conservative  to  state  that 
the  first  decade  and  a  half  of  the  present  century  (twentieth) 
produced  at  least  one-fifth  as  much  (mathematical  literature) 
as  all  the  preceding  centuries  combined.  Hence  it  appears 
likely  that  the  twentieth  will  produce,  as  the  nineteenth  century 
has  done,  much  more  new  mathematical  literature  than  the 
total  existing  mathematical  literature  at  the  beginning.’ ’ 

C.  J.  Keyser,  in  a  lecture  ten  years  ago,  “2,000  books  and 
memoirs  drop  from  the  mathematical  press  of  the  world  in  a 
single  year,  the  estimated  number  amounting  to  50,000  in  the 
last  generation.  .  .  . 

“Indeed,  the  modern  developments  of  mathematics  consti¬ 
tutes  not  only  one  of  the  most  impressive,  but  one  of  the  most 
characteristic,  phenomena  of  our  age.  It  is  a  phenomenon, 
however,  of  which  the  boasted  intelligence  of  ‘universalized’ 
daily  press  seems  strangely  unaware ;  and  there  is  no  other  great 
human  interest,  whether  of  science  or  of  art,  regarding  which  the 
mind  of  the  educated  public  is  permitted  to  hold  so  many 
fallacious  opinions  and  inferior  estimates.  The  golden  age  of 
mathematics — that  was  not  the  age  of  Euclid,  it  is  ours.” 

President  N.  M.  Butler,  of  Columbia  University,  “Modern 
mathematics,  that  most  astonishing  of  intellectual  creation, 
has  projected  the  mind’s  eye  through  infinite  time  and  the 
mind’s  hand  into  boundless  space.” 

Jas.  Pierpont,  “Surely  this  is  the  golden  age  of  mathematics.” 


ALFRED  DAVIS, 

Chairman, 

Francis  W.  Parker  School, 
330  Webster  Ave.,  Chicago,  Ill. 

J.  A.  FOBERG, 

Crane  Technical  High  School, 

Chicago,  Ill. 
A  M.  ALLISON, 

Lakeview  High  School, 

Chicago,  Ill. 


M.  J.  NEWELL, 

Evanston  High  School, 

Evanston,  Ill. 
C.  M.  AUSTIN, 

Oak  Park  High  School, 

Oak  Park,  Ill. 
J.  R.  CLARK,  . 

President  Ex-Officio, 
Parker  High  School, 

Chicago,  ill. 


